Mathematics
Directions (Q. 49 to 52): Answer these questions on the basis of the following information:
Through the mid-point M of the side CD of a ∥gm ABCD, the line BM is drawn, intersecting AC on L and AD produced in E.
49. Which of the following is true for triangles BMC and EMD ?
I. They are similar to each other.
II. They are congruent to each other.
III. Their areas are equal.
IV. Their perimeters are equal.
I and II
II and III
I and III
I, II, III and IV
50. ΔAEL is similar to:
ΔCBL
ΔCML
ΔDME
ΔBMC
51. By which axiom are the above triangles similar?
AA
SSS
SAS
ASA
52. EL : BL is equal to :
1 : 2
2 : 1
1 : 3
3 : 1
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Answer
49. Given,
In ΔBMC and ΔEMD,
∠EDM = ∠MCB [Alternate angles are equal]
∠EMD = ∠BMC [Vertically opposite angles are equal]
CM = DM (As M is the mid-point of CD)
ΔBMC ≅ ΔEMD [By A.A. axiom]
Congruent triangles are similar, have equal areas and perimeters.
∴ All statements are correct
Hence, option 4 is the correct option.
50. Given,
In ΔAEL and ΔCBL,
∠EAL = ∠BCL[Alternate angles are equal]
∠ALE = ∠CLB [Vertical opposite angles are equal]
ΔAEL ∼ ΔCBL [By A.A. axiom]
Hence, option 1 is the correct option.
51. Given,
The similarity ΔAEL ∼ ΔCBL was established by using two pairs of corresponding angles.
Hence, option 1 is the correct option.
52. Given,
AD = BC [ABCD is a parallelogram]
BC = DE [ΔBMC ≅ ΔEMD]
From figure,
AE = AD + DE
AE = BC + BC
AE = 2BC
Using ΔAEL ∼ ΔCBL,
Since, corresponding sides of similar triangle are proportional to each other.
Hence, option 2 is the correct option.
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Related Questions
Directions (Q. 41 to 44): These questions are based on the following information :
In the given figure, ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. It is given that AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 cm and PQ = 2 cm.
41. The perimeter of ΔABC is:
12 cm
16 cm
18 cm
24 cm
42. The ratio of the areas of ΔAPQ and ΔABC is:
1 : 3
1 : 4
1 : 9
1 : 16
43. Which of the following holds true?
ΔAPQ ∼ ΔABC
ΔAQP ∼ ΔABC
ΔAPQ ∼ ΔACB
none of these
44. Which axiom of similarity applies in the above case?
AAA
AA
SSS
SAS
Directions (Q. 45 to 48): Using the given diagram answer the following questions.
In ΔPQR, AB ∥ QR, QP ∥ CB and AR intersects CB at O.
45. The triangle similar to ΔARQ is:
ΔORC
ΔARP
ΔOBR
ΔQRP
46. ΔPQR ∼ ΔBCR by axiom:
SAS
AAA
SSS
AAS
47. If QC = 6 cm, CR = 4 cm, BR = 3 cm, then the length of RP is:
4.5 cm
5 cm
7.5 cm
8 cm
48. The ratio PQ : BC is:
2 : 3
3 : 2
2 : 5
5 : 2
Directions (Q. 53 to 56): Study the following diagram carefully and answer the given questions:
In ΔABC, D and E are points on AB and AC respectively such that AD = a, DB = 3a, AE = b and EC = 3b. DQ ∥ EA and EP ∥ DA are drawn. QP is joined.
53. ΔADE is similar to which of the following triangles?
I. ΔABC
II. ΔDAQ
III. ΔADQ
IV. ΔEPA
V. ΔEAP
I, II and IV only
I, III and V only
I, II and V only
I, III and IV only
54. If DE = 2 cm, then BC is equal to:
4 cm
6 cm
7 cm
8 cm
55. The ratio of the perimeters of ΔADE and ΔABC is:
1 : 2
1 : 3
1 : 4
1 : 6
56. The ratio of the areas of ΔADE and trapezium DBCE is:
1 : 8
1 : 9
1 : 15
1 : 16
Directions (Q. 57 to 60): Study the given information and answer the questions that follow:
In the given figure ABCD is a trapezium in which DC is parallel to AB. AB = 16 cm and DC = 8 cm, OD = 5 cm, OB = (y + 3) cm, OA = 11 cm and OC = (x − 1) cm.
57. From the given figure name the pair of similar triangles :
ΔAOD, ΔOBC
ΔCOD, ΔAOB
ΔADB, ΔACB
ΔCOD, ΔCOB
58. The corresponding proportional sides with respect to the pair of similar triangles obtained above is :
59. The ratio of the sides of the pair of similar triangles is:
1 : 3
1 : 2
2 : 3
3 : 1
60. Using the ratio of sides of the pair of similar triangles, the values of x and y are respectively :
x = 4.6, y = 7
x = 7, y = 7
x = 6.5, y = 7
x = 6.5, y = 2