Similarity of Triangles

In the given figure, XY || BC. Given that AX = 3 cm, XB = 1.5 cm and BC = 6 cm.

(i) Calculate $\dfrac{AY}{YC}$.

(ii) Calculate XY.

Answer

(i) By basic proportionality theorem,

A line drawn parallel to a side of triangle divides the other two sides proportionally.

Since, XY || BC

$\therefore \dfrac{AY}{YC} = \dfrac{AX}{XB} \\[1em] = \dfrac{3}{1.5} \\[1em] = \dfrac{2}{1}.$

Hence, $\dfrac{AY}{YC} = \dfrac{2}{1}$.

(ii) In ΔAXY and ΔABC,

∠AXY = ∠ABC [Corresponding angles are equal]

∠XAY = ∠BAC [Common ]

∴ ΔAXY ∼ ΔABC.

Since, corresponding sides of similar triangle are proportional to each other.

$\Rightarrow \dfrac{AX}{AB} = \dfrac{XY}{BC} \\[1em] \Rightarrow \dfrac{AX}{AX + XB} = \dfrac{XY}{BC} \\[1em] \Rightarrow \dfrac{3}{3 + 1.5} = \dfrac{XY}{6} \\[1em] \Rightarrow \dfrac{3}{4.5} = \dfrac{XY}{6} \\[1em] \Rightarrow XY = \dfrac{3}{4.5} \times 6 \\[1em] \Rightarrow XY = 4 \text{ cm.}$

Hence, XY = 4 cm.

In the given figure, DE || BC.

(i) If AD = 3.6 cm, AB = 9 cm and AE = 2.4 cm, find EC.

(ii) If $\dfrac{AD}{DB} = \dfrac{3}{5}$ and AC = 5.6 cm, find AE.

(iii) If AD = x cm, DB = (x − 2) cm, AE = (x + 2) cm and EC = (x − 1) cm, find the value of x.

Answer

By basic proportionality theorem,

A line drawn parallel to a side of triangle divides the other two sides proportionally.

(i) Given,

AD = 3.6 cm

AB = 9 cm

AE = 2.4 cm

Since, DE || BC by basic proportionality theorem,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{AD}{AB - AD} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{3.6}{9 - 3.6} = \dfrac{2.4}{EC}\\[1em] \Rightarrow \dfrac{3.6}{5.4} = \dfrac{2.4}{EC}\\[1em] \Rightarrow EC = \dfrac{5.4 \times 2.4}{3.6}\\[1em] \Rightarrow EC = 3.6 \text{ cm.}$

Hence, EC = 3.6 cm.

(ii) Given,

$\dfrac{AD}{DB} = \dfrac{3}{5}$

AC = 5.6 cm

Since, DE || BC by basic proportionality theorem,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{AC - AE} \\[1em] \Rightarrow \dfrac{3}{5} = \dfrac{AE}{5.6 - AE}\\[1em] \Rightarrow 3 \times (5.6 - AE) = 5AE \\[1em] \Rightarrow 16.8 - 3AE = 5AE \\[1em] \Rightarrow 5AE + 3AE = 16.8 \\[1em] \Rightarrow 8AE = 16.8 \\[1em] \Rightarrow AE = \dfrac{16.8}{8} \\[1em] \Rightarrow AE = 2.1 \text{ cm.}$

Hence, AE = 2.1 cm.

(iii) Given,

AD = x cm

DB = (x − 2) cm

AE = (x + 2) cm

EC = (x − 1) cm

Since, DE || BC by basic proportionality theorem,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{x}{(x - 2)} = \dfrac{(x + 2)}{(x - 1)} \\[1em] \Rightarrow x(x - 1) = (x + 2) (x - 2) \\[1em] \Rightarrow x^2 - x = (x^2 - (2)^2) \\[1em] \Rightarrow x^2 - x = x^2 - 4 \\[1em] \Rightarrow x^2 - x - x^2 + 4 = 0 \\[1em] \Rightarrow - x + 4 = 0 \\[1em] \Rightarrow x = 4 \text{ cm.}$

Hence, x = 4 cm.

D and E are points on the sides AB and AC respectively of ΔABC. For each of the following cases, state whether DE ∥ BC :

(i) AD = 5.7 cm, BD = 9.5 cm, AE = 3.6 cm and EC = 6 cm.

(ii) AB = 5.6 cm, AD = 1.4 cm, AC = 9.6 cm and EC = 2.4 cm.

(iii) AB = 11.7 cm, BD = 5.2 cm, AE = 4.4 cm and AC = 9.9 cm.

(iv) AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm and AE = 2.8 cm.

Answer

By basic proportionality theorem,

A line drawn parallel to a side of triangle divides the other two sides proportionally.

(i) Given,

AD = 5.7 cm

BD = 9.5 cm

AE = 3.6 cm

EC = 6 cm.

Check for proportionality,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{5.7}{9.5} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{3}{5} = 0.6 \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{3.6}{6.0} \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{3}{5} = 0.6 \\[1em] \therefore \dfrac{AD}{DB} = \dfrac{AE}{EC}.$

We conclude that DE is parallel to BC

Hence, DE is parallel to BC.

(ii) Given,

AB = 5.6 cm

AD = 1.4 cm

AC = 9.6 cm

EC = 2.4 cm.

From figure,

DB = AB - AD = 5.6 - 1.4 = 4.2 cm

AE = AC - EC = 9.6 - 2.4 = 7.2 cm

Check for proportionality,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{1.4}{4.2} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{1}{3} \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{7.2}{2.4} \\[1em] \Rightarrow \dfrac{AE}{EC} = 3 \\[1em] \therefore \dfrac{AD}{DB} \ne \dfrac{AE}{EC}.$

We conclude that DE is not parallel to BC

Hence, DE is not parallel to BC.

(iii) Given,

AB = 11.7 cm

BD = 5.2 cm

AE = 4.4 cm

AC = 9.9 cm.

From figure,

AD = AB - BD = 11.7 - 5.2 = 6.5 cm

EC = AC - AE = 9.9 - 4.4 = 5.5 cm

Check for proportionality,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{6.5}{5.2} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{5}{4} = 1.25 \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{4.4}{5.5} \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{4}{5} = 0.8 \\[1em] \therefore \dfrac{AD}{DB} \ne \dfrac{AE}{EC}.$

We conclude that DE is not parallel to BC

Hence, DE is not parallel to BC.

(iv) Given,

AB = 10.8 cm

BD = 4.5 cm

AC = 4.8 cm

AE = 2.8 cm.

AD = AB - BD = 10.8 - 4.5 = 6.3 cm

EC = AC - AE = 4.8 - 2.8 = 2 cm

Check for proportionality,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{6.3}{4.5} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{7}{5} = 1.4 \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{2.8}{2.0} \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{7}{5} = 1.4 \\[1em] \therefore \dfrac{AD}{DB} = \dfrac{AE}{EC}.$

We conclude that DE is parallel to BC

Hence, DE is parallel to BC.

In the given figure, it is given that ∠ABD = ∠CDB = ∠PQB = 90°. If AB = x units, CD = y units and PQ = z units, prove that $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z}$.

Answer

In ΔPQD and ΔABD,

∠ABD = ∠PQD = 90° [From figure]

∠ADB = ∠PDQ [Common angles]

∴ ΔPQD ∼ ΔABD (By A.A. axiom)

Corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{PQ}{AB} = \dfrac{QD}{BD} \\[1em] \Rightarrow \dfrac{z}{x} = \dfrac{QD}{BD} \text{ .....(1)}$

In ΔPQB and ΔCDB,

∠CDB = ∠PQB = 90° [Given]

∠CBD = ∠PBQ [Common angles]

∴ ΔPQB ∼ ΔCDB (By A.A. axiom)

Corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{PQ}{CD} = \dfrac{BQ}{BD} \\[1em] \Rightarrow \dfrac{z}{y} = \dfrac{BQ}{BD} \text{ ....(2)}$

Add equations (1) and (2) we get,

$\Rightarrow \dfrac{z}{x} + \dfrac{z}{y} = \dfrac{QD}{BD} + \dfrac{BQ}{BD} \\[1em] \Rightarrow z \Big(\dfrac{1}{x} + \dfrac{1}{y}\Big) = \dfrac{BD}{BD} \\[1em] \Rightarrow z \Big(\dfrac{1}{x} + \dfrac{1}{y}\Big) = 1 \\[1em] \Rightarrow \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z}.$

Hence, proved that $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z}$.

In ΔABC, AD is the bisector of ∠A. If BC = 10 cm, BD = 6 cm and AC = 6 cm, find AB.

Answer

Construction: Draw a line through C parallel to AD, meeting BA produced at E.

In ΔBCE,

Since AD ∥ CE, by Basic Proportionality Theorem:

$\dfrac{BD}{DC} = \dfrac{AB}{AE}$ ...(i)

Also, since AD ∥ CE:

∠BAD = ∠AEC [Corresponding angles are equal]

∠DAC = ∠ACE [Alternate interior angles are equal]

Since AD is bisector of ∠A, ∠BAD = ∠DAC.

Therefore, ∠AEC = ∠ACE.

In ΔACE, sides opposite to equal angles are equal:

AE = AC ...(ii)

Substitute (ii) into (i):

$\dfrac{BD}{DC} = \dfrac{AB}{AC}$

Given,

AC = 6 cm

BC = 10 cm

BD = 6 cm

DC = BC - BD [from figure]

DC = 10 - 6 = 4 cm

Let length of AB be x,

$\Rightarrow \dfrac{BD}{DC} = \dfrac{AB}{AC} \\[1em] \Rightarrow \dfrac{6}{4} = \dfrac{x}{6} \\[1em] \Rightarrow \dfrac{3}{2} = \dfrac{x}{6} \\[1em] \Rightarrow \dfrac{3}{2} \times 6 = x \\[1em] \Rightarrow x = 9.$

Hence, length of AB is 9 cm.

In the given figure, AC ∥ DE ∥ BF. If AC = 24 cm, EG = 8 cm, GB = 16 cm, BF = 30 cm.

(i) Prove that ΔGED ∼ ΔGBF.

(ii) Find DE.

(iii) Find DB : AB.

Answer

(i) In ΔGED and ΔGBF,

∠DGE = ∠BGF [Vertically opposite angles are equal]

∠GED = ∠GBF [Alternate angles are equal]

∴ ΔGED ∼ ΔGBF (By A.A. axiom)

Hence, proved ΔGED ∼ ΔGBF.

(ii) We know that,

Corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{DE}{BF} = \dfrac{EG}{GB} \\[1em] \Rightarrow \dfrac{DE}{30} = \dfrac{8}{16} \\[1em] \Rightarrow \dfrac{DE}{30} = \dfrac{1}{2} \\[1em] \Rightarrow DE = \dfrac{1}{2} \times 30 \\[1em] \Rightarrow DE = 15 \text{ cm}.$

Hence, DE = 15 cm.

(iii) In ΔDBE and ΔABC,

∠EBD = ∠CBA [Common angles]

∠BDE = ∠BAC [Corresponding angles are equal, Since AC ∥ DE]

ΔDBE ∼ ΔABC (By A.A. axiom)

Corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{DB}{AB} = \dfrac{DE}{AC} \\[1em] \Rightarrow \dfrac{DB}{AB} = \dfrac{15}{24} \\[1em] \Rightarrow \dfrac{DB}{AB} = \dfrac{5}{8}.$

Hence, DB : AB = 5 : 8.

In the adjoining figure, ABCD is a parallelogram, P is a point on side BC and DP when produced meets AB produced at L. Prove that :

(i) DP : PL = DC : BL.

(ii) DL : DP = AL : DC.

Answer

(i) Given,

ABCD is a parallelogram.

∴ AB || DC and AD || BC

In ΔDPC and ΔLPB,

∠PDC = ∠PLB [Alternate angles are equal]

∠DPC = ∠LPB [Vertically opposite angles are equal]

∴ ΔDPC ∼ ΔLPB (By A.A. axiom)

Corresponding sides of similar triangles are proportional.

$\dfrac{DP}{LP} = \dfrac{DC}{LB}$

DP : PL = DC : BL

Hence, proved DP : PL = DC : BL.

(ii) In ΔALD and ΔCPD,

∠ALD = ∠PDC [Alternate interior angles, AB || DC]

∠DAL = ∠PCD [Opposite angles of a parallelogram are equal]

∴ ΔLAD ∼ ΔDCP (By A.A. axiom)

Corresponding sides of similar triangles are proportional.

$\dfrac{DL}{DP} = \dfrac{AL}{DC}$

DL : DP = AL : DC

Hence, proved DL : DP = AL : DC.

In the given figure, ABCD is a parallelogram, E is a point on BC and the diagonal BD intersects AE at F.

Prove that DF × FE = FB × FA.

Answer

Since, ABCD is a || gm

∴ AD || BC

In ΔADF and ΔEBF,

∠ADF = ∠EBF [Alternate angles are equal]

∠AFD = ∠EFB [Vertically opposite angles are equal]

∴ ΔADF ∼ ΔEBF (By A.A. axiom)

Corresponding sides of similar triangles are proportional.

$\dfrac{DF}{FB} = \dfrac{FA}{FE}$

DF × FE = FB × FA

Hence, proved that DF × FE = FB × FA.

In the adjoining figure (not drawn to scale), PS = 4 cm, SR = 2 cm, PT = 3 cm and QT = 5 cm.

(i) Show that ΔPQR ∼ ΔPST.

(ii) Calculate ST, if QR = 5.8 cm.

Answer

(i) Given,

PS = 4 cm, SR = 2 cm, PT = 3 cm and QT = 5 cm.

PR = PS + SR = 4 + 2 = 6 cm

PQ = PT + TQ = 3 + 5 = 8 cm

In ΔPQR and ΔPST,

$\Rightarrow \dfrac{PQ}{PS} = \dfrac{8}{4} = 2 \\[1em] \Rightarrow \dfrac{PR}{PT} = \dfrac{6}{3} = 2 \\[1em] \therefore \dfrac{PQ}{PS} = \dfrac{PR}{PT}$

∠QPR = ∠SPT [Common angle]

∴ ΔPQR ∼ ΔPST [By S.A.S. axiom]

Hence, ΔPQR ∼ ΔPST.

(ii) We know that,

Corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{QR}{ST} = \dfrac{PQ}{PS} \\[1em] \Rightarrow \dfrac{5.8}{ST} = \dfrac{8}{4} \\[1em] \Rightarrow ST = \dfrac{5.8}{2} \\[1em] \Rightarrow ST = 2.9 \text{ cm.}$

Hence, ST = 2.9 cm.

In the adjoining figure, ABCD is a parallelogram in which AB = 16 cm, BC = 10 cm and L is a point on AC such that CL : LA = 2 : 3. If BL produced meets CD at M and AD produced at N, prove that :

(i) ΔCLB ∼ ΔALN.

(ii) ΔCLM ∼ ΔALB.

Answer

(i) In ΔCLB and ΔALN,

∠CBL = ∠ANL [Alternate angles are equal]

∠CLB = ∠ALN [Vertically opposite angles are equal]

∴ ΔCLB ∼ ΔALN (By A.A. axiom)

Hence, proved that ΔCLB ∼ ΔALN.

(ii) In ΔCLM and ΔALB,

∠LMC = ∠LAB [Alternate angles, since CM ∥ AB, transversal LM]

∠CLM = ∠ALB [Vertically opposite angles are equal]

∴ ΔCLM ∼ ΔALB (By A.A. axiom)

Hence, proved that ΔCLM ∼ ΔALB.

In the given figure, AB ∥ PQ and AC ∥ PR. Prove that BC ∥ QR.

Answer

In ΔOPQ,

AB ∥ PQ [Given]

By Basic Proportionality Theorem,

$\dfrac{OA}{AP} = \dfrac{OB}{BQ}$ .... (i)

In ΔOPR,

AC ∥ PR [Given]

By Basic Proportionality Theorem,

$\dfrac{OA}{AP} = \dfrac{OC}{CR}$ .... (ii)

From (i) and (ii), we get:

$\dfrac{OB}{BQ} = \dfrac{OC}{CR}$

In ΔOQR,

Since $\dfrac{OB}{BQ} = \dfrac{OC}{CR}$

By Converse of Basic Proportionality Theorem,

BC ∥ QR

Hence, proved that BC ∥ QR.

In the given figure, medians AD and BE of ΔABC meet at G and DF ∥ BE. Prove that :

(i) EF = FC.

(ii) AG : GD = 2 : 1.

Answer

(i) In ∆CFD and ∆CEB,

∠CDF = ∠CBE [Corresponding angles are equal]

∠FCD = ∠ECB [Common]

∴ ∆CFD ∼ ∆CEB (By A.A. axiom)

Since, corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{CF}{CE} = \dfrac{CD}{CB} \\[1em] \Rightarrow \dfrac{CF}{CE} = \dfrac{1}{2} \text{ [∵ D is the mid-point of BC]} \Rightarrow CE = 2CF.$

From figure,

⇒ CE = CF + FE

⇒ 2CF = CF + FE

⇒ CF = FE.

Hence, proved that EF = FC.

(ii) In ∆AFD,

GE ∥ DF

By basic proportionality theorem we have,

$\dfrac{AE}{EF} = \dfrac{AG}{GD}$ .....(1)

Now, AE = EC [∵ BE is media,n so E is the mid-point of AC]

As, AE = EC = 2EF [As, EF = FC].

Substituting value of AE in (1) we get,

$\dfrac{AG}{GD} = \dfrac{2EF}{EF} = \dfrac{2}{1}.$

Hence, proved that AG : GD = 2 : 1.

In the given figure, DE ∥ BC and BD = DC.

(i) Prove that DE bisects ∠ADC.

(ii) If AD = 4.5 cm, AE = 3.9 cm and DC = 7.5 cm, find CE.

(iii) Find the ratio AD : DB.

Answer

(i) Given, DE ∥ BC.

⇒ ∠ADE = ∠DBC [Corresponding angles are equal] ... (1)

⇒ ∠EDC = ∠DCB [Alternate interior angles are equal] ... (2)

Given, BD = DC.

⇒ ∠DBC = ∠DCB [Angles opposite to equal sides are equal] ... (3)

From (1), (2), and (3), we get:

∠ADE = ∠EDC

Thus, DE bisects ∠ADC.

Hence, DE bisects ∠ADC.

(ii) Given,

AD = 4.5 cm, AE = 3.9 cm, DC = 7.5 cm

Given,

BD = DC = 7.5 cm

By basic proportionality theorem we have,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{4.5}{7.5} = \dfrac{3.9}{EC} \\[1em] \Rightarrow 4.5 \times EC = 3.9 \times 7.5 \\[1em] \Rightarrow 4.5 \times EC = 29.25 \\[1em] \Rightarrow EC = \dfrac{29.25}{4.5} \\[1em] \Rightarrow EC = 6.5 \text{ cm.}$

Hence, CE = 6.5 cm.

(iii) By basic proportionality theorem,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{3.9}{6.5} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{3}{5}.$

Hence, the ratio AD : DB is 3 : 5.

In the given figure, BA ∥ DC. Show that ΔOAB ∼ ΔODC. If AB = 4 cm, CD = 3 cm, OC = 5.7 cm and OD = 3.6 cm, find OA and OB.

Answer

Considering ΔAOB and ΔDOC,

∠AOB = ∠COD [Vertically opposite angles are equal]

∠A = ∠D [Alternate angles are equal]

∴ ΔAOB ∼ ΔDOC (By A.A. axiom)

We know that,

Corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{OA}{OD} = \dfrac{OB}{OC} = \dfrac{AB}{DC}$

Considering,

$\Rightarrow \dfrac{AB}{DC} = \dfrac{OA}{OD} \\[1em] \Rightarrow \dfrac{4}{3} = \dfrac{OA}{3.6} \\[1em] \Rightarrow OA = \dfrac{4 \times 3.6}{3} \\[1em] \Rightarrow OA = \dfrac{14.4}{3} \\[1em] \Rightarrow OA = 4.8 \text{ cm}.$

Considering,

$\Rightarrow \dfrac{AB}{DC} = \dfrac{OB}{OC} \\[1em] \Rightarrow \dfrac{4}{3} = \dfrac{OB}{5.7} \\[1em] \Rightarrow OB = \dfrac{4 \times 5.7}{3} \\[1em] \Rightarrow OB = \dfrac{22.8}{3} \\[1em] \Rightarrow OB = 7.6 \text{ cm}.$

Hence, OA = 4.8 cm and OB = 7.6 cm.

In the given figure, ∠ABC = 90° and BD ⟂ AC. If AB = 5.7 cm, BD = 3.8 cm and CD = 5.4 cm, find BC.

Answer

We have, ∠ABC = 90° and BD ⟂ AC

In ΔABC and ΔBDC,

∠ABC = ∠BDC [Each 90°]

∠ACB = ∠BCD [Common]

∴ ΔABC ∼ ΔBDC (By A.A. axiom)

We know that,

Corresponding sides of similar triangles are proportional.

$\therefore \dfrac{AB}{BD} = \dfrac{BC}{DC} \\[1em] \Rightarrow \dfrac{5.7}{3.8} = \dfrac{BC}{5.4} \\[1em] \Rightarrow BC = \dfrac{5.7 \times 5.4}{3.8}\\[1em] \Rightarrow BC = \dfrac{30.78}{3.8}\\[1em] \Rightarrow BC = 8.1 \text{ cm}.$

Hence, BC = 8.1 cm.

In the given figure, DE ∥ BC.

(i) Prove that ΔADE ∼ ΔABC.

(ii) Given that AD = $\dfrac{1}{2}$ DB, calculate DE, if BC = 4.5 cm.

(iii) Find $\dfrac{\text{ar(ΔADE)}}{\text{ar(ΔABC)}}$.

(iv) Find $\dfrac{\text{ar(ΔADE)}}{\text{ar(trap. BCED)}}$.

Answer

(i) Considering ΔADE and ΔABC,

∠BAC = ∠DAE [Common angles]

∠ADE = ∠ABC [Corresponding angles are equal]

∴ ΔADE ∼ ΔABC (By A.A. axiom)

Hence, proved that ΔADE ∼ ΔABC.

(ii) Given,

$\Rightarrow AD = \dfrac{1}{2} BD \\[1em] \Rightarrow AD = \dfrac{1}{2}(AB - AD) \\[1em] \Rightarrow 2AD = AB - AD \\[1em] \Rightarrow 2AD + AD = AB \\[1em] \Rightarrow 3AD = AB \\[1em] \Rightarrow \dfrac{AD}{AB} = \dfrac{1}{3} \\[1em] \Rightarrow AD : AB = 1 : 3.$

Since triangles ADE and ABC are similar so, ratio of their corresponding sides will be equal.

$\therefore \dfrac{AD}{AB} = \dfrac{DE}{BC} \\[1em] \therefore \dfrac{1}{3} = \dfrac{DE}{4.5} \\[1em] \therefore DE = \dfrac{4.5}{3} \\[1em] \therefore DE = 1.5 \text{ cm}.$

Hence, the length of DE = 1.5 cm.

(iii) We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\Rightarrow \dfrac{\text{ar(ΔADE)}}{\text{ar(ΔABC)}} = \dfrac{DE^2}{BC^2} \\[1em] = \Big(\dfrac{DE}{BC}\Big)^2 \\[1em] = \Big(\dfrac{1}{3}\Big)^2 \\[1em] = \dfrac{1}{9}.$

Hence, $\dfrac{\text{ar(ΔADE)}}{\text{ar(ΔABC)}} = \dfrac{1}{9}$.

(iv) From figure,

Area of trap.(BCED) = area(Δ ABC) - area(Δ ADE)

$\Rightarrow \dfrac{\text{ar(ΔABC)}}{\text{ar(ΔADE)}} = \dfrac{9}{1} \\[1em] \Rightarrow \text{ar(ΔABC)} = 9[\text{ar(ΔADE)}]\\[1em] \Rightarrow \text{ar(trap. BCED)} = 9[\text{ar(ΔADE)}] - \text{ar(ΔADE)} \\[1em] \Rightarrow \text{ar(trap. BCED)} = 8[\text{ar(ΔADE)}] \\[1em] \Rightarrow \dfrac{\text{ar(ΔADE)}}{\text{ar(trap. BCED)}} = \dfrac{1}{8}.$

Hence, $\dfrac{\text{ar(ΔADE)}}{\text{ar(trap. BCED)}} = \dfrac{1}{8}$.

Given that ΔABC ∼ ΔPQR.

(i) If ar(ΔABC) = 49 cm² and ar(ΔPQR) = 25 cm² and AB = 5.6 cm, find the length of PQ.

(ii) If ar(ΔABC) = 28 cm² and ar(ΔPQR) = 63 cm² and PR = 8.4 cm, find the length of AC.

(iii) If BC = 4 cm, QR = 5 cm and ar(ΔABC) = 32 cm² determine ar(ΔPQR).

Answer

(i) Given,

ΔABC ∼ ΔPQR

ar(ΔABC) = 49 cm²

ar(ΔPQR) = 25 cm²

AB = 5.6 cm

We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\therefore \dfrac{\text{ar(ΔABC)}}{\text{ar(ΔPQR)}} = \Big(\dfrac{AB}{PQ}\Big)^2 \\[1em] \Rightarrow \dfrac{49}{25} = \Big(\dfrac{5.6}{PQ}\Big)^2 \\[1em] \Rightarrow \sqrt{\dfrac{49}{25}} = \dfrac{5.6}{PQ} \\[1em] \Rightarrow \dfrac{7}{5} = \dfrac{5.6}{PQ} \\[1em] \Rightarrow PQ = \dfrac{5 \times 5.6}{7}\\[1em] \Rightarrow PQ = \dfrac{28}{7}\\[1em] \Rightarrow PQ = 4 \text{ cm.}$

Hence, PQ = 4 cm.

(ii) Given,

ΔABC ∼ ΔPQR

ar(ΔABC) = 28 cm²

ar(ΔPQR) = 63 cm²

PR = 8.4 cm

We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\therefore \dfrac{\text{ar(ΔABC)}}{\text{ar(ΔPQR)}} = \Big(\dfrac{AC}{PR}\Big)^2 \\[1em] \Rightarrow \dfrac{28}{63} = \Big(\dfrac{AC}{8.4}\Big)^2 \\[1em] \Rightarrow \sqrt{\dfrac{4}{9}} = \dfrac{AC}{8.4} \\[1em] \Rightarrow \dfrac{2}{3} = \dfrac{AC}{8.4} \\[1em] \Rightarrow AC = \dfrac{2 \times 8.4}{3}\\[1em] \Rightarrow AC = \dfrac{16.8}{3}\\[1em] \Rightarrow AC = 5.6 \text{ cm.}$

Hence, AC = 5.6 cm.

(iii) Given,

ΔABC ∼ ΔPQR

BC = 4 cm

QR = 5 cm

ar(ΔABC) = 32 cm²

We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\therefore \dfrac{\text{ar(ΔABC)}}{\text{ar(ΔPQR)}} = \Big(\dfrac{BC}{QR}\Big)^2 \\[1em] \Rightarrow \dfrac{32}{\text{ar(ΔPQR)}} = \Big(\dfrac{4}{5}\Big)^2 \\[1em] \Rightarrow \dfrac{32}{\text{ar(ΔPQR)}} = \dfrac{16}{25} \\[1em] \Rightarrow \text{ar(ΔPQR)} = \dfrac{32 \times 25}{16} \\[1em] \Rightarrow \text{ar(ΔPQR)} = 25 \times 2 \\[1em] \Rightarrow \text{ar(ΔPQR)} = 50 \text{ cm}^2.$

Hence, ar(ΔPQR) = 50 cm².

The areas of two similar triangles are 48 cm² and 75 cm² respectively. If the altitude of the first triangle is 3.6 cm, find the corresponding altitude of the other.

Answer

Let the length of altitude of other triangle be x cm

We know that, the ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding altitudes.

$\therefore \dfrac{\text{Area of first Δ}}{\text{Area of second Δ}} = \Big(\dfrac{\text{Altitude of first Δ}}{\text{Altitude of second Δ}}\Big)^2 \\[1em] \Rightarrow \dfrac{48}{75} = \Big(\dfrac{3.6}{x}\Big)^2 \\[1em] \Rightarrow \dfrac{16}{25} = \Big(\dfrac{3.6}{x}\Big)^2 \\[1em] \Rightarrow \sqrt{\dfrac{16}{25}} = \dfrac{3.6}{x} \\[1em] \Rightarrow \dfrac{4}{5} = \dfrac{3.6}{x} \\[1em] \Rightarrow x = \dfrac{18}{4} \\[1em] \Rightarrow x = 4.5 \text{ cm.}$

Hence, the length of altitude of other triangle = 4.5 cm.

In the given figure, AB ⟂ BC and DE ⟂ BC. If AB = 9 cm, DE = 3 cm and AC = 24 cm, calculate AD.

Answer

Given,

AB ⟂ BC and DE ⟂ BC

Thus AB ∥ DE.

∠ABC = ∠DEC [Given]

∠ACB = ∠DCE [Common angle in both triangles]

∴ ΔABC ∼ ΔDEC (By A.A. axiom)

From figure,

DC = AC - AD = 24 - AD

We know that,

Corresponding sides of similar triangles are proportional.

$\therefore \dfrac{DE}{AB} = \dfrac{DC}{AC} \\[1em] \Rightarrow \dfrac{3}{9} = \dfrac{24 - AD}{24} \\[1em] \Rightarrow 24 - AD = \dfrac{3 \times 24}{9} \\[1em] \Rightarrow 24 - AD = \dfrac{72}{9} \\[1em] \Rightarrow 24 - AD = 8 \\[1em] \Rightarrow AD = 24 - 8 \\[1em] \Rightarrow AD = 16 \text{ cm.}$

Hence, AD = 16 cm.

In the given figure, DE || BC. If DE = 4 cm, BC = 6 cm and ar(ΔADE) = 20 cm², find the area of ΔABC.

Answer

Considering ΔADE and ΔABC,

∠A = ∠A [Common angles]

∠ADE = ∠ABC [Corresponding angles are equal]

∴ ΔADE ∼ ΔABC (By A.A. axiom)

We know that,

The ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides.

Let the area of ΔABC be x cm².

$\therefore \dfrac{\text{Area of ΔADE}}{\text{Area of ΔABC}} = \Big(\dfrac{{DE}}{{BC}}\Big)^2 \\[1em] \Rightarrow \dfrac{\text{Area of ΔADE}}{\text{Area of ΔABC}} = \dfrac{{DE}^2}{{BC}^2} \\[1em] \Rightarrow \dfrac{20}{x} = \dfrac{4^2}{6^2} \\[1em] \Rightarrow \dfrac{20}{x} = \dfrac{16}{36} \\[1em] \Rightarrow x = \dfrac{20 \times 36}{16} \\[1em] \Rightarrow x = \dfrac{720}{16} \\[1em] \Rightarrow x = 45 \text{ cm}^2$

Hence, area of ΔABC = 45 cm².

In the given figure, LM ∥ BC. If AB = 6 cm, AL = 2 cm and AC = 9 cm, calculate :

(i) the length of CM,

(ii) Find the value of $\dfrac{\text{ar(ΔALM)}}{\text{ar(trap. LBCM)}}$.

Answer

(i) Given,

AB = 6 cm

AL = 2 cm

LB = AB - AL = 6 - 2 = 4 cm

AC = AM + MC

AM = AC - MC

AM = 9 - MC

In ΔAML and ΔABC,

∠AML = ∠ACB [Corresponding angles are equal]

∠LAM = ∠BAC [Common angle]

ΔAML ∼ ΔABC [By AA similarity]

We know that,

Corresponding sides of similar triangles are proportional.

$\therefore \dfrac{AL}{LB} = \dfrac{AM}{MC} \\[1em] \Rightarrow \dfrac{2}{4} = \dfrac{9 - MC}{MC} \\[1em] \Rightarrow 2MC = 4(9 - MC) \\[1em] \Rightarrow 2MC = 36 - 4MC \\[1em] \Rightarrow 2MC + 4MC = 36 \\[1em] \Rightarrow 6MC = 36 \\[1em] \Rightarrow MC = \dfrac{36}{6} \\[1em] \Rightarrow MC = 6 \text{ cm.}$

Hence, CM = 6 cm.

(ii) Given

In ΔALM and ΔABC,

∠AML = ∠ACB [Corresponding angles are equal]

∠LAM = ∠BAC [Common angle]

ΔAML ∼ ΔABC [By AA similarity]

We know that,

The ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides.

$\therefore \dfrac{\text{ar(ΔALM)}}{\text{ar(ΔABC)}} = \Big(\dfrac{AL}{AB}\Big)^2 \\[1em] \Rightarrow \dfrac{\text{ar(ΔALM)}}{\text{ar(ΔABC)}} = \Big(\dfrac{2}{6}\Big)^2 \\[1em] \Rightarrow \dfrac{\text{ar(ΔALM)}}{\text{ar(ΔABC)}} = \dfrac{4}{36} \\[1em] \Rightarrow \dfrac{\text{ar(ΔALM)}}{\text{ar(ΔABC)}} = \dfrac{1}{9}.$

Let ar(ΔALM) = x, then ar(ΔABC) = 9x.

From figure,

ar(trap. LBCM) = ar(ΔABC) - ar(ΔALM)

= 9x - x

= 8x.

$\Rightarrow \dfrac{\text{ar(ΔALM)}}{\text{ar(trap. LBCM)}} = \dfrac{x}{8x} = \dfrac{1}{8}$.

Hence, $\dfrac{\text{ar(ΔALM)}}{\text{ar(trap. LBCM)}} = \dfrac{1}{8}$.

In ΔABC, it is given that AB = 12 cm, ∠B = 90° and AC = 15 cm. If D and E are points on AB and AC respectively such that ∠AED = 90° and DE = 3 cm, prove that :

(i) ΔABC ∼ ΔAED.

(ii) ar(ΔAED) = 6 cm².

(iii) ar(quad BCED) : ar(ΔABC) = 8 : 9.

Answer

(i) Given,

∠ABC = ∠AED = 90° [Given]

∠BAC = ∠DAE [Common angle]

∴ ΔABC ∼ ΔAED (By A.A. axiom)

Hence, proved that ΔABC ∼ ΔAED.

(ii) ΔABC is right-angled triangle, applying pythagoras theorem,

⇒ AC² = AB² + BC²

⇒ BC² = AC² - AB²

⇒ BC² = (15)² - (12)²

⇒ BC² = 225 - 144

⇒ BC² = 81

⇒ BC = $\sqrt{81}$

⇒ BC = 9 cm

Area of ΔABC = $\dfrac{1}{2}$ × Base × height

= $\dfrac{1}{2}$ × 12 × 9

= 54 cm²

Since, ΔABC ∼ ΔAED,

We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\therefore \dfrac{\text{ar(ΔAED)}}{\text{ar(ΔABC)}} = \dfrac{{ED}^2}{{BC}^2} \\[1em] \Rightarrow \dfrac{\text{ar(ΔAED)}}{\text{ar(ΔABC)}} = \dfrac{3^2}{9^2} \\[1em] \Rightarrow \dfrac{\text{ar(ΔAED)}}{\text{ar(ΔABC)}} = \dfrac{9}{81} \\[1em] \Rightarrow \dfrac{\text{ar(ΔAED)}}{\text{ar(ΔABC)}} = \dfrac{1}{9} \\[1em] \Rightarrow \text{ar(ΔAED)} = \text{ar(ΔABC)} \times \dfrac{1}{9} \\[1em] \Rightarrow \text{ar(ΔAED)} = \dfrac{54}{9} \\[1em] \Rightarrow \text{ar(ΔAED)} = 6 \text{ cm}^2$

Hence, proved that ar(ΔAED) = 6 cm².

(iii) From figure,

ar(quad. BCED) = ar(ΔABC) - ar(ΔAED)

ar(quad. BCED) = 54 - 6

ar(quad. BCED) = 48 cm².

$\Rightarrow \dfrac{\text{ar(quad. BCED)}}{\text{ar.(ΔABC)}} = \dfrac{48}{54} = \dfrac{8}{9}$.

Hence, proved that ar(quad BCED) : ar(ΔABC) = 8 : 9.

In the given figure, ∠PQR = ∠PST = 90°, PQ = 5 cm and PS = 2 cm.

(i) Prove that ΔPQR ∼ ΔPST.

(ii) Find area of ΔPQR : Area of quadrilateral SRQT.

Answer

(i) Considering ΔPQR and ΔPST.

∠P = ∠P [Common angles]

∠PQR = ∠PST [Both are equal to 90°]

∴ ΔPQR ∼ ΔPST (By A.A. axiom)

Hence, proved that ΔPQR ∼ ΔPST.

(ii) We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\Rightarrow \dfrac{\text{Area of ΔPQR}}{\text{Area of ΔPST}} = \dfrac{{PQ}^2}{{PS}^2} \\[1em] \Rightarrow \dfrac{\text{Area of ΔPQR}}{\text{Area of ΔPST}} = \dfrac{5^2}{2^2} \\[1em] \Rightarrow \dfrac{\text{Area of ΔPQR}}{\text{Area of ΔPQR} - \text{Area of SRQT}} = \dfrac{25}{4} \\[1em] \Rightarrow 4\text{ Area of ΔPQR} = 25 (\text{ Area of ΔPQR} - \text{Area of SRQT}) \\[1em] \Rightarrow 4\text{ Area of ΔPQR} = 25 \text{ Area of ΔPQR} - 25\text{Area of SRQT} \\[1em] \Rightarrow 25\text{Area of SRQT} = 25 \text{ Area of ΔPQR} - 4\text{ Area of ΔPQR}\\[1em] \Rightarrow 25\text{Area of SRQT} = 21 \text{ Area of ΔPQR} \\[1em] \Rightarrow \dfrac{\text{Area of ΔPQR}}{\text{Area of SRQT}} = \dfrac{25}{21}.$

Hence, area of ΔPQR : Area of quadrilateral SRQT = 25 : 21.

In a ΔPQR, L and M are two points on the base QR such that ∠LPQ = ∠RQP and ∠RPM = ∠RQP. Prove that

(i) ΔPQL ∼ ΔRPM.

(ii) QL × RM = PL × PM.

(iii) PQ² = QL × QR.

Answer

(i) In ΔPQL and ΔRPM

∠LPQ = ∠MRP [Given]

∠LQP = ∠RPM [Given]

∴ ΔPQL ∼ ΔRPM (y A.A. axiom)

Hence, proved that ΔPQL ∼ ΔRPM.

(ii) Since, ΔPQL ∼ ΔRPM and corresponding sides of similar triangle are proportional to each other.

$\therefore \dfrac{QL}{PM} = \dfrac{PL}{RM} \\[1em] \Rightarrow QL \times RM = PL \times PM.$

Hence, proved that QL × RM = PL × PM.

(iii) In ΔPQL and ΔRQP

∠LPQ = ∠QRP [Given ]

∠Q = ∠Q [Common]

∴ ΔPQL ∼ ΔRQP (By A.A. axiom)

Since, corresponding sides of similar triangle are proportional to each other.

$\therefore \dfrac{PQ}{RQ} = \dfrac{QL}{QP} \\[1em] \Rightarrow PQ^2 = QR \times QL.$

Hence, proved that PQ² = QR x QL.

In the adjoining figure, the medians BD and CE of a ΔABC meet at G. Prove that :

(i) ΔEGD ∼ ΔCGB.

(ii) BG = 2 × GD.

Answer

(i) Since, BD and CE are medians.

So, E is mid-point of AB and D is mid-point of AC.

By mid-point theorem,

ED ∥ BC and ED = $\dfrac{1}{2}$ BC [By mid-point theorem]

$\Rightarrow \dfrac{ED}{BC} = \dfrac{1}{2}$

$\Rightarrow \dfrac{BC}{ED} = 2$ .....(1)

In triangle EGD and BGC,

∠EGD = ∠BGC [Vertically opposite angles are equal]

∠DEG = ∠GCB [Alternate angles are equal]

∴ ΔEGD ∼ ΔCGB by (By A.A. axiom)

Hence, proved that ΔEGD ∼ ΔCGB.

(ii) Since, corresponding sides of similar triangle are proportional to each other.

$\Rightarrow \dfrac{BG}{GD} = \dfrac{BC}{ED} \\[1em] \Rightarrow \dfrac{BG}{GD} = 2 .....\text{(From 1)} \\[1em] \Rightarrow BG = 2GD.$

Hence, proved that BG = 2GD.

In the adjoining figure, PQRS is a parallelogram with PQ = 15 cm and RQ = 10 cm. If L is a point on RP such that RL : PL = 2 : 3 and QL produced meets RS at M and PS produced at N, find the lengths of PN and RM.

Answer

In ΔRLQ and ΔPLN,

⇒ ∠RLQ = ∠PLN [Vertically opposite angles are equal]

⇒ ∠LRQ = ∠LPN [Alternate angles are equal]

∴ ΔRLQ ∼ ΔPLN (By A.A. axiom)

Since, corresponding sides of similar triangles are proportional we have :

$\Rightarrow \dfrac{RL}{LP} = \dfrac{RQ}{PN} \\[1em] \Rightarrow \dfrac{2}{3} = \dfrac{10}{PN} \\[1em] \Rightarrow PN = \dfrac{30}{2} = 15 \text{ cm}.$

In ΔRLM and ΔPLQ,

⇒ ∠RLM = ∠PLQ [Vertically opposite angles are equal]

⇒ ∠LRM = ∠LPQ [Alternate angles are equal]

∴ ΔRLM ∼ ΔPLQ (By A.A. axiom)

Since, corresponding sides of similar triangles are proportional we have :

$\Rightarrow \dfrac{RM}{PQ} = \dfrac{RL}{LP} \\[1em] \Rightarrow \dfrac{RM}{15} = \dfrac{2}{3} \\[1em] \Rightarrow RM = \dfrac{30}{3} \\[1em] \Rightarrow RM = 10 \text{ cm.}$

Hence, PN = 15 cm and RM = 10 cm.

In the given figure, ΔABC ∼ ΔPQR, AM and PN are altitudes, whereas AX and PY are medians. Prove that $\dfrac{AM}{PN} = \dfrac{AX}{PY}$.

Answer

Since ΔABC ∼ ΔPQR

So, their respective sides will be in proportion

$\dfrac{AB}{PQ} = \dfrac{AC}{PR} = \dfrac{BC}{QR}$

Also, ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R

In ΔABM and ΔPQN,

∠ABM = ∠PQN [Since, ABC and PQR are similar]

∠AMB = ∠PNQ = 90° [Given ]

∴ ΔΑΒΜ ∼ ΔPQN by AA similarity

$\dfrac{AM}{PN} = \dfrac{AB}{PQ}$ .....(1)

Since, AX and PY are medians so they will divide their opposite sides.

BX = $\dfrac{BC}{2}$ and QY = $\dfrac{QR}{2}$

Therefore, we have:

$\dfrac{AB}{PQ} = \dfrac{BX}{QY}$

∠ABC = ∠PQR

So, we had observed that two respective sides are in same proportion in both triangles and also angle included between them is respectively equal.

Hence, ∆ABX ∼ ∆PQY (by SAS similarity rule).So,

$\dfrac{AM}{PQ} = \dfrac{AX}{PY}$ .....(2)

From (1) and (2),

$\dfrac{AM}{PN} = \dfrac{AX}{PY}$

Hence, proved that $\dfrac{AM}{PN} = \dfrac{AX}{PY}$

In the given figure, BC ∥ DE, area (ΔABC) = 25 cm², area (trap. BCED) = 24 cm² and DE = 14 cm. Calculate the length of BC.

Answer

Area of ΔADE = Area of ΔABC + Area of trapezium BCED = 25 + 24 = 49 cm².

Given,

BC ∥ DE.

In ΔABC and ΔADE,

∠ABC = ∠ADE [Corresponding angles are equal]

∠ACB = ∠AED [Corresponding angles are equal]

∴ ΔABC ∼ ΔADE by (By A.A. axiom)

We know that,

The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides.

$\therefore \dfrac{\text{Area of ΔABC}}{\text{Area of ΔADE}} = \dfrac{BC^2}{DE^2} \\[1em] \Rightarrow \dfrac{25}{49} = \dfrac{BC^2}{14^2} \\[1em] \Rightarrow BC^2 = \dfrac{25}{49} \times 196 \\[1em] \Rightarrow BC^2 = 100 \\[1em] \Rightarrow BC = 10 \text{ cm.}$

Hence, BC = 10 cm.

In the given figure, DE ∥ BC and DE : BC = 3 : 5. alculate ar(ΔADE) : ar(trap. BCED).

Answer

It is given that DE ∥ BC.

∠ADE = ∠ABC [Corresponding angles are equal]

∠AED = ∠ACB [Corresponding angles are equal]

∴ ΔADE ∼ ΔАВС (By A.A. axiom)

We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\dfrac{\text{ar(ΔABC)}}{\text{ar(ΔADE)}} = \dfrac{(BC)^2}{(DE)^2}$

Subtracting 1 from both sides, we get:

$\Rightarrow \dfrac{\text{ar(ΔABC)}}{\text{ar(ΔADE)}} - 1 = \dfrac{(5)^2}{(3)^2} - 1 \\[1em] \Rightarrow \dfrac{\text{ar(ΔABC)} - \text{ar(ΔADE)}}{\text{ar(ΔADE)}} = \dfrac{25}{9} - 1 \\[1em] \Rightarrow \dfrac{\text{ar(BCED)}}{\text{ar(ΔADE)}} = \dfrac{25 - 9}{9} \\[1em] \Rightarrow \dfrac{\text{ar(BCED)}}{\text{ar(ΔADE)}} = \dfrac{16}{9} \\[1em] \Rightarrow \dfrac{\text{ar(ΔADE)}}{\text{ar(trap. BCED)}} = \dfrac{9}{16}.$

Hence, ar(ΔADE) : ar(trap. BCED) = 9 : 16.

In ΔABC, D and E are mid-points of AB and AC respectively.
Find: ar(ΔADE) : ar(ΔABC).

Answer

We know that,

The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of the third side.

Given,

In Δ ABC,

D is mid-point of side AB and E is the mid-point of the side AC.

∴ DE ∥ BC and DE = $\dfrac{1}{2}$ BC

D is mid-point of side AB.

∴ AB = 2AD

Let us consider ΔADE and ΔABC

∠DAE = ∠BAC [Common angle]

∠ADE = ∠ABC [Corresponding angle are equal]

∴ ΔADE ∼ ΔABC by AA similarity.

$\Rightarrow \dfrac{\text{area(ΔADE)}}{\text{area(ΔABC)}} = \dfrac{AD^2}{AB^2} \\[1em] \Rightarrow \dfrac{\text{area(ΔADE)}}{\text{area(ΔABC)}} = \dfrac{AD^2}{(2AD)^2} \\[1em] \Rightarrow \dfrac{\text{area(ΔADE)}}{\text{area(ΔABC)}} = \dfrac{1}{4}.$

Hence, ar(ΔADE) : ar(ΔABC) = 1 : 4.

In ΔPQR, MN is parallel to QR and $\dfrac{PM}{QM} = \dfrac{2}{3}$.

(i) Find $\dfrac{MN}{QR}$.

(ii) Prove that ΔOMN and ΔORQ are similar.

(iii) Find: Area of ΔOMN : Area of ΔORQ.

Answer

(i) Considering ΔPMN and ΔPQR,

∠P = ∠P [Common angles]

∠PMN = ∠PQR [Corresponding angles are equal]

∴ ΔPMN ∼ ΔPQR by AA similarity.

Given,

$\dfrac{PM}{MQ} = \dfrac{2}{3} \\[1em] \Rightarrow \dfrac{PM}{PQ - PM} = \dfrac{2}{3} \\[1em] \Rightarrow 3PM = 2(PQ - PM) \\[1em] \Rightarrow 3PM = 2PQ - 2PM \\[1em] \Rightarrow 3PM + 2PM = 2PQ \\[1em] \Rightarrow 5PM = 2PQ \\[1em] \Rightarrow \dfrac{PM}{PQ} = \dfrac{2}{5}.$

Since triangles are similar hence the ratio of the corresponding sides will be equal,

$\dfrac{MN}{QR} = \dfrac{PM}{PQ} = \dfrac{2}{5}$.

Hence, $\dfrac{MN}{QR} = \dfrac{2}{5}$

(ii) Considering ΔOMN and ΔORQ,

∠MON = ∠QOR (Vertically opposite angles are equal)

∠OMN = ∠ORQ (Alternate angles are equal)

Hence, by AA similarity ΔOMN ∼ ΔORQ.

(iii) We know that, the ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides.

$\therefore \dfrac{\text{Area of ΔOMN}}{\text{Area of ΔORQ}} = \dfrac{MN^2}{QR^2} \\[1em] \Rightarrow \dfrac{\text{Area of ΔOMN}}{\text{Area of ΔORQ}} = \dfrac{2^2}{5^2} \\[1em] \Rightarrow \dfrac{\text{Area of ΔOMN}}{\text{Area of ΔORQ}} = \dfrac{4}{25} \\[1em]$

Hence, the ratio of the Area of ΔOMN : Area of ΔORQ = 4 : 25.

PQR is a triangle, S is a point on the side QR of ΔPQR such that ∠PSR = ∠QPR. Given QP = 8 cm, PR = 6 cm and SR = 3 cm.

(i) Prove ΔPQR ∼ ΔSPR.

(ii) Find the length of QR and PS.

(iii) Find $\dfrac{\text{area of ΔPQR}}{\text{area of ΔSPR}}$.

Answer

(i) In ΔPQR and ΔSPR,

⇒ ∠PSR = ∠QPR [Given ]

⇒ ∠PRQ = ∠PRS [Common angle]

∴ ΔPQR ∼ ΔSPR by AA similarity.

Hence, proved that ΔPQR ∼ ΔSPR.

(ii) Since, ΔPQR ∼ ΔSPR and corresponding sides of similar triangle are proportional to each other.

$\Rightarrow \dfrac{QR}{PR} = \dfrac{PR}{SR} \\[1em] \Rightarrow \dfrac{QR}{6} = \dfrac{6}{3} \\[1em] \Rightarrow \dfrac{QR}{6} = \dfrac{36}{3} = 12 \text{ cm.}$

Also,

$\Rightarrow \dfrac{PQ}{SP} = \dfrac{PR}{SR} \\[1em] \Rightarrow \dfrac{8}{SP} = \dfrac{6}{3} \\[1em] \Rightarrow SP = \dfrac{8 \times 6}{3} \\[1em] \Rightarrow SP = \dfrac{24}{3} = 4 \text{ cm.}$

Hence, QR = 12 cm and PS = 4 cm.

(iii) We know that,

Ratio of areas of two similar triangles is same as the square of the ratio between their corresponding sides.

$\therefore \dfrac{\text{Area of ΔPQR}}{\text{Area of ΔPQR}} = \Big(\dfrac{PQ}{SP}\Big)^2 \\[1em] = \Big(\dfrac{8}{4}\Big)^2 \\[1em] = (2)^2 \\[1em] = 4$

Hence, $\dfrac{\text{Area of ΔPQR}}{\text{Area of ΔPQR}} = \dfrac{4}{1}$

If ΔABC and ΔDEF are similar triangles in which ∠A = 47° and ∠E = 83°, then ∠C equals :

1. 50°

2. 60°

3. 70°

4. 80°

Answer

Given,

ΔABC ∼ ΔDEF.

Thus, corresponding angles are equal.

∠B = ∠E = 83°

We know that, sum of all angles in triangle is equal to 180°.

∠A + ∠B + ∠C = 180°

47° + 83° + ∠C = 180°

130° + ∠C = 180°

∠C = 180° - 130°

∠C = 50°.

Hence, option 1 is the correct option.

If ΔABC ∼ ΔQRP, then the corresponding proportional sides are :

1. $\dfrac{AB}{PQ} = \dfrac{BC}{RP}$

2. $\dfrac{AB}{QR} = \dfrac{BC}{QP}$

3. $\dfrac{AC}{QR} = \dfrac{BC}{RP}$

4. $\dfrac{AB}{QR} = \dfrac{BC}{RP}$

Answer

Given,

ΔABC ∼ ΔQRP.

$\therefore \dfrac{AB}{QR} = \dfrac{BC}{RP}$

Hence, option 4 is the correct option.

If in ΔABC and ΔPQR, we have $\dfrac{AB}{QR} = \dfrac{BC}{PR} = \dfrac{CA}{PQ}$, then:

1. ΔPQR ∼ ΔCAB

2. ΔPQR ∼ ΔABC

3. ΔBCA ∼ ΔPQR

4. ΔCBA ∼ ΔPQR

Answer

Given,

The two triangles are ΔABC and ΔPQR.

$\dfrac{AB}{QR} = \dfrac{BC}{PR} = \dfrac{CA}{PQ}$

By SSS similarity criterion, if the sides of one triangle are proportional to the sides of another triangle, then their corresponding angles are equal and the triangles are similar.

To find the correspondence of vertices:

The side AB corresponds to QR. The vertex opposite to AB is C, and opposite to QR is P.

C = P

The side BC corresponds to PR. The vertex opposite to BC is A, and opposite to PR is Q.

A = Q

The side CA corresponds to PQ. The vertex opposite to CA is B, and opposite to PQ is R.

B = R

∴ΔPQR ∼ ΔCAB

Hence, option 1 is the correct option.

In the given figure, AB = 24 cm, AC = 18 cm, DE = 12 cm, DF = 9 cm and ∠BAC = ∠EDF. Then ΔABC ∼ ΔEDF by the condition :

1. AAA

2. SAS

3. SSS

4. AAS

Answer

$\dfrac{AB}{DE} = \dfrac{24}{12} = 2$

$\dfrac{AC}{DF} = \dfrac{18}{9} = 2$

Thus,

$\dfrac{AB}{DE} = \dfrac{AC}{DF}$

∠BAC = ∠EDF [Given ]

∴ ΔABC ∼ ΔEDF (By S.A.S. axiom)

Hence, option 2 is the correct option.

If ΔABC and ΔDEF are so related that $\dfrac{AB}{FD} = \dfrac{BC}{DE} = \dfrac{CA}{EF}$, then which of the following is true?

1. ∠A = ∠E and ∠B = ∠D

2. ∠B = ∠F and ∠C = ∠D

3. ∠A = ∠F and ∠B = ∠D

4. ∠C = ∠F and ∠A = ∠D

Answer

Given,

$\dfrac{AB}{FD} = \dfrac{BC}{DE} = \dfrac{CA}{EF}$

∴ ΔABC ∼ ΔDEF BY SSS theorem.

Side AB corresponds to side FD. Therefore, vertex A corresponds to vertex F and vertex B corresponds to vertex D.

Side BC corresponds to side DE. Therefore, vertex B corresponding to D. It also implies vertex C corresponds to vertex E. 

Since the triangles are similar, the corresponding angles are also similar,

∠A = ∠F and ∠B = ∠D

Hence, option 3 is the correct option.

In the given figure, PQ is parallel to TR, then by using condition of similarity:

1. $\dfrac{PQ}{RT} = \dfrac{OP}{OT} = \dfrac{OQ}{OR}$

2. $\dfrac{PQ}{RT} = \dfrac{OP}{OR} = \dfrac{OQ}{OT}$

3. $\dfrac{PQ}{RT} = \dfrac{OR}{OP} = \dfrac{OQ}{OT}$

4. $\dfrac{PQ}{RT} = \dfrac{OP}{OR} = \dfrac{OT}{OQ}$

Answer

Given,

ΔPOQ and ΔROT

∠POQ = ∠TOR [vertically opposite angles are equal]

∠OPQ = ∠ORT [Alternate interior angles are equal]

∴ ΔPOQ ∼ ΔROT (By A.A. axiom)

We know that,

The ratio of corresponding sides in similar triangles are proportional.

$\dfrac{PQ}{RT} = \dfrac{OP}{OR} = \dfrac{OQ}{OT}$

Hence, option 2 is the correct option.

D and E are points on the sides AB and AC respectively of ΔABC. In which of the following cases DE ∥ BC?

1. AD = 3 cm, BD = 8 cm, AC = 8 cm, AE = 3 cm

2. AD = 5 cm, BD = 6 cm, AE = 6 cm, CE = 5 cm

3. AB = 18 cm, AD = 8 cm, AE = 12 cm, EC = 15 cm

4. none of these

Answer

Checking option 3,

Given,

In ΔABC, DE is parallel to BC if and only if it divides the sides AB and AC in the same ratio

From figure,

DB = AB - AD = 18 - 8 = 10 cm

$\dfrac{AD}{DB} = \dfrac{8}{10} = \dfrac{4}{5}$

$\dfrac{AE}{EC} = \dfrac{12}{15} = \dfrac{4}{5}$

$\therefore \dfrac{AD}{DB} = \dfrac{AE}{EC}$

AB = 18 cm, AD = 8 cm, AE = 12 cm, EC = 15 cm DE is parallel to BC holds true.

Hence, option 3 is the correct option.

In ΔABC, DE ∥ BC such that $\dfrac{AD}{DB} = \dfrac{3}{5}$. If AC = 5.6 cm, then AE = ?

1. 2.1 cm

2. 2.8 cm

3. 3.1 cm

4. 4.2 cm

Answer

By basic proportionality theorem,

If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those two sides in the same ratio.

Since DE ∥ BC in ΔABC,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{3}{5} = \dfrac{AE}{AC - AE} \\[1em] \Rightarrow \dfrac{3}{5} = \dfrac{AE}{5.6 - AE} \\[1em] \Rightarrow 3 \times (5.6 - AE) = 5AE \\[1em] \Rightarrow 16.8 - 3AE = 5AE \\[1em] \Rightarrow 5AE + 3AE = 16.8 \\[1em] \Rightarrow 8AE = 16.8 \\[1em] \Rightarrow AE = \dfrac{16.8}{8} \\[1em] \Rightarrow AE = 2.1 \text{ cm.}$

Hence, option 1 is the correct option.

In ΔABC, DE ∥ BC so that AD = (7x − 4) cm, AE = (5x − 2) cm, DB = (3x + 4) cm and EC = (3x) cm.
Then x equals:

1. 2.5

2. 3

3. 4

4. 5

Answer

By basic proportionality theorem,

If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those two sides in the same ratio.

Since DE ∥ BC in ΔABC,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{(7x - 4)}{(3x + 4)} = \dfrac{(5x - 2)}{3x} \\[1em] \Rightarrow 3x \times (7x - 4) = (5x - 2) \times (3x + 4) \\[1em] \Rightarrow 21x^2 - 12x = 15x^2 + 20x -6x -8 \\[1em] \Rightarrow 21x^2 - 12x - 15x^2 - 20x + 6x + 8 = 0\\[1em] \Rightarrow 21x^2 - 15x^2 - 12x - 20x + 6x + 8 = 0 \\[1em] \Rightarrow 6x^2 - 26x + 8 = 0 \\[1em] \Rightarrow 6x^2 - 24x - 2x + 8 = 0 \\[1em] \Rightarrow 6x(x - 4) - 2(x - 4) = 0 \\[1em] \Rightarrow (6x - 2)(x - 4) = 0 \\[1em] \Rightarrow (6x - 2) = 0 \text{ or } (x - 4) = 0 \text{[Using Zero-product rule]}\\[1em] \Rightarrow 6x= 2 \text{ or } x = 4 \\[1em] \Rightarrow x= \dfrac{2}{6} \text{ or } x = 4 \\[1em] \Rightarrow x= \dfrac{1}{3} \text{ or } x = 4 \\[1em]$

x cannot be $\dfrac{1}{3}$ as it yields negative AD and AE values. Therefore x = 4.

Hence, option 3 is the correct option.

It is given that ΔABC ∼ ΔDFE. If ∠A = 30°, ∠C = 50°, AB = 5 cm, AC = 8 cm and DF = 7.5 cm, then which of the following is true?

1. DE = 12 cm, ∠F = 50°

2. DE = 12 cm, ∠F = 100°

3. EF = 12 cm, ∠D = 100°

4. EF = 12 cm, ∠D = 30°

Answer

Given,

∠A = 30°

∠C = 50°

In triangle ABC,

∠A + ∠B + ∠C = 180°

∠B = 180° - (∠A + ∠C)

∠B = 180° - (30° + 50°)

∠B = 180° - 80°

∠B = 100°.

As the triangles are similar, the corresponding angles will be equal.

So, ∠F = ∠B = 100°

Since the triangles are similar, the ratio of corresponding sides is proportional,

$\Rightarrow \dfrac{AB}{DF} = \dfrac{AC}{DE} = \dfrac{BC}{FE} \\[1em] \text{Let's consider} \\[1em] \Rightarrow \dfrac{AB}{DF} = \dfrac{AC}{DE} \\[1em] \Rightarrow \dfrac{5}{7.5} = \dfrac{8}{DE} \\[1em] \Rightarrow DE = \dfrac{8 \times 7.5}{5} \\[1em] \Rightarrow DE = \dfrac{60}{5} \\[1em] \Rightarrow DE = 12 \text{ cm.}$

DE = 12 cm, ∠F = 100°

Hence, option 2 is the correct option.

In ΔDEF and ΔPQR, it is given that ∠D = ∠Q and ∠R = ∠E, then which of the following is not true?

1. $\dfrac{DE}{PQ} = \dfrac{EF}{RP}$

2. $\dfrac{DE}{QR} = \dfrac{DF}{PQ}$

3. $\dfrac{EF}{PR} = \dfrac{DF}{PQ}$

4. $\dfrac{EF}{RP} = \dfrac{DE}{QR}$

Answer

Given, in ΔDEF and ΔPQR:

∠D = ∠Q

∠E = ∠R

∴ ΔDEF ∼ ΔQRP by AA similarity.

The ratio of corresponding sides is:

$\dfrac{DE}{QR} = \dfrac{DF}{QP}$

$\dfrac{DE}{PQ} = \dfrac{EF}{RP}$ do not hold true.

Hence, option 1 is the correct option.

In ΔABC, D and E are points on AB and AC respectively such that DE ∥ BC. If AE = 2 cm, EC = 3 cm and BC = 10 cm, then DE is equal to:

1. 4 cm

2. 5 cm

3. $\dfrac{20}{3}$ cm

4. 15 cm

Answer

From figure,

AC = AE + EC = 2 + 3 = 5 cm.

In ΔABC and ΔADE,

∠A = ∠A [Common angle]

∠ABC = ∠ADE [Corresponding angles are equal]

∴ ΔABC ∼ ΔADE (By A.A. axiom).

In similar triangles, ratios of corresponding sides are equal.

$\therefore \dfrac{AE}{AC} = \dfrac{DE}{BC} \\[1em] \Rightarrow \dfrac{2}{5} = \dfrac{DE}{10} \\[1em] \Rightarrow DE = \dfrac{2 \times 10}{5} \\[1em] \Rightarrow DE = \dfrac{20}{5} \\[1em] \Rightarrow DE = 4 \text{ cm.}$

Hence, option 1 is the correct option.

ΔABC is such that AB = 3 cm, BC = 2 cm and CA = 2.5 cm. ΔDEF ∼ ΔABC. If EF = 4 cm, then the perimeter of ΔDEF is :

1. 7.5 cm

2. 15 cm

3. 22.5 cm

4. 30 cm

Answer

Given,

ΔDEF ∼ ΔABC

Perimeter of ΔABC = AB + BC + CA = 3 + 2 + 2.5 = 7.5 cm

Since the two triangles are similar, we can write:

$\Rightarrow \dfrac{\text{Perimeter of ΔABC}}{\text{Perimeter of ΔDEF}} = \dfrac{BC}{EF} \\[1em] \Rightarrow \dfrac{7.5}{\text{Perimeter of ΔDEF}} = \dfrac{2}{4} \\[1em] \Rightarrow \text{Perimeter of ΔDEF} = \dfrac{7.5 \times 4}{2} \\[1em] \Rightarrow \text{Perimeter of ΔDEF} = \dfrac{30}{2} \\[1em] \Rightarrow \text{Perimeter of ΔDEF} = 15 \text{ cm.}$

Hence, option 2 is the correct option.

In ΔABC, DE is drawn parallel to BC cutting the other two sides at D and E. If AB = 3.6 cm, AC = 2.4 cm and AD = 2.1 cm, then AE is equal to:

1. 1.05 cm

2. 1.2 cm

3. 1.4 cm

4. 1.8 cm

Answer

Given,

In ΔABC and ΔADE,

∠A = ∠A [Common angle]

∠ABC = ∠ADE [Corresponding angles are equal]

∴ ΔABC ∼ ΔADE (By A.A. axiom).

We know that,

Corresponding sides of similar triangle are proportional.

$\Rightarrow \dfrac{AB}{AD} = \dfrac{AC}{AE} \\[1em] \Rightarrow \dfrac{3.6}{2.1} = \dfrac{2.4}{AE} \\[1em] \Rightarrow AE = \dfrac{2.4 \times 2.1}{3.6} \\[1em] \Rightarrow AE = \dfrac{5.04}{3.6} \\[1em] \Rightarrow AE = 1.4 \text{ cm.}$

Hence, option 3 is the correct option.

In the given figure, ∠BAP = ∠DCP = 70°, PC = 6 cm and CA = 4 cm, then PD : DB is equal to:

1. 5 : 3

2. 3 : 5

3. 3 : 2

4. 2 : 3

Answer

Given,

∠BAP = ∠DCP = 70°

From figure,

∠BAP and ∠DCP are corresponding angles and since they are equal.

∴ AB ∥ CD.

By basic proportionality theorm,

$\Rightarrow \dfrac{PD}{DB} = \dfrac{PC}{CA} \\[1em] \Rightarrow \dfrac{PD}{DB} = \dfrac{6}{4} \\[1em] \Rightarrow \dfrac{PD}{DB} = \dfrac{3}{2} \\[1em] \Rightarrow PD : DB = 3 : 2.$

Hence, option 3 is the correct option.

In the adjoining figure, DE ∥ BC. If AD : DB = 3 : 1 and EA = 3.3 cm, then AC equals:

1. 1.1 cm

2. 4 cm

3. 4.4 cm

4. 5.5 cm

Answer

Since, DE // BC,

By basic proportionality theorem,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{3}{1} = \dfrac{3.3}{EC} \\[1em] \Rightarrow EC = \dfrac{3.3}{3} \\[1em] \Rightarrow EC = 1.1 \text{ cm.}$

From figure,

AC = AE + EC

AC = 3.3 + 1.1 = 4.4 cm.

Hence, option 3 is the correct option.

In the adjoining figure, ∠ADE = ∠ABC, AE = 8 cm, EB = 7 cm, BC = 9 cm, AD = 10 cm and DC = 2 cm. Then the length of DE is:

1. 6 cm

2. 6.75 cm

3. 7.8 cm

4. 13.5 cm

Answer

From figure,

AB = AE + EB = 8 + 7 = 15 cm

AC = AD + DC = 10 + 2 = 12 cm

In ΔABC and ΔADE

∠BAC = ∠DAE [Common angle]

∠ABC = ∠ADE [Given]

∴ ΔABC ∼ ΔADE (By A.A. axiom).

We know that,

In similar triangles the ratio of corresponding sides are equal.

$\Rightarrow \dfrac{AD}{AB} = \dfrac{AE}{AC} = \dfrac{DE}{BC} \\[1em] \text{Let's Consider} \\[1em] \Rightarrow \dfrac{AD}{AB} = \dfrac{DE}{BC} \\[1em] \Rightarrow \dfrac{10}{15} = \dfrac{DE}{9} \\[1em] \Rightarrow DE = \dfrac{10 \times 9}{15} \\[1em] \Rightarrow DE = \dfrac{90}{15} \\[1em] \Rightarrow DE = 6 \text{ cm.}$

Hence, option 1 is the correct option.

In ΔABC, it is given that AB = 9 cm, BC = 6 cm and CA = 7.5 cm. Also ΔDEF is given such that EF = 8 cm and ΔDEF ∼ ΔABC. Then the perimeter of ΔDEF is:

1. 22.5 cm

2. 25 cm

3. 27 cm

4. 30 cm

Answer

Given,

ΔDEF ∼ ΔABC

Perimeter of ΔABC = AB + BC + CA = 9 + 6 + 7.5 = 22.5 cm.

Since the triangles are similar,

$\therefore \dfrac{\text{Perimeter of ΔABC}}{\text{Perimeter of ΔDEF}} = \dfrac{BC}{EF} \\[1em] \Rightarrow \dfrac{22.5}{\text{Perimeter of ΔDEF}} = \dfrac{6}{8} \\[1em] \Rightarrow \text{Perimeter of ΔDEF} = \dfrac{22.5 \times 8}{6} \\[1em] \Rightarrow \text{Perimeter of ΔDEF} = \dfrac{180}{6} \\[1em] \Rightarrow \text{Perimeter of ΔDEF} = 30 \text{ cm.}$

Hence, option 4 is the correct option.

In a ΔABC, AB = 10 cm, AC = 14 cm and BC = 6 cm. If AD is the internal bisector of ∠A, then CD is equal to:

1. 3.5 cm

2. 4.8 cm

3. 7 cm

4. 10.5 cm

Answer

Construction: Draw a line through C parallel to AD, meeting BA produced at E.

Since AD ∥ EC and BE is the transversal:

∠BAD = ∠AEC [Corresponding angles are equal]

Since AD ∥ EC and AC is the transversal:

∠DAC = ∠ACE [Alternate interior angles]

Given AD is the bisector of ∠A:

∠BAD = ∠DAC

∴ ∠AEC = ∠ACE

In ΔACE, sides opposite to equal angles are equal.

∴ AE = AC = 14 cm

Let DC be x,

Now, in ΔBCE, we have AD ∥ EC. By Basic Proportionality Theorem,

$\Rightarrow \dfrac{BD}{DC} = \dfrac{AB}{AE} \\[1em] \Rightarrow \dfrac{BD}{DC} = \dfrac{10}{14} \\[1em] \Rightarrow \dfrac{BD}{DC} = \dfrac{10}{14} \\[1em] \Rightarrow \dfrac{BD}{DC} = \dfrac{5}{7} \\[1em]$

Let CD = x.

Then BD = BC - CD = 6 - x.

Substituting these values into the ratio:

$\Rightarrow \dfrac{6 - x}{x} = \dfrac{5}{7} \\[1em] \Rightarrow 7(6 - x) = 5x \\[1em] \Rightarrow 42 - 7x = 5x \\[1em] \Rightarrow 42 = 5x + 7x \\[1em] \Rightarrow 12x = 42 \\[1em] \Rightarrow x = \dfrac{42}{12} \\[1em] \Rightarrow x = 3.5 \text{ cm.}$

Thus, CD = 3.5 cm.

Hence, option 1 is the correct option.

In the given figure, two line segments AC and BD intersect each other at point P such that PA = 6 cm, PB = 3 cm, PC = 2.5 cm, PD = 5 cm, ∠APB = 50° and ∠CDP = 30°. Then ∠PBA = ?

1. 30°

2. 50°

3. 60°

4. 100°

Answer

Considering ΔAPB and ΔCPD,

$\dfrac{AP}{PD} = \dfrac{6}{5} \text{ and } \dfrac{BP}{CP} = \dfrac{3}{2.5} = \dfrac{6}{5}$