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Mathematics

Assertion (A): In the figure, ∠ABC = ∠BDC = 90°.
If AD = 4 cm, BD = 6 cm, then area of ΔABC is 40 cm2.

Reason (R): Areas of two similar triangles are proportional to the squares of their corresponding sides.

  1. A is true, R is false

  2. A is false, R is true

  3. Both A and R are true

  4. Both A and R are false

Areas of two similar triangles are proportional to the squares of their corresponding sides. Similarity of Triangles, RSA Mathematics Solutions ICSE Class 10.

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Answer

In ΔADB and ΔBDC,

∠ADB = ∠BDC = 90°

∠DBA = ∠DCB [Angles complementary to ∠DBC]

∴ ΔADB ∼ ΔBDC [By A.A. axiom]

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

⇒ADBD=BDDC⇒46=6DC⇒4×DC=36⇒DC=364⇒DC=9 cm.\Rightarrow \dfrac{AD}{BD} = \dfrac{BD}{DC} \\[1em] \Rightarrow \dfrac{4}{6} = \dfrac{6}{DC} \\[1em] \Rightarrow 4 \times DC = 36 \\[1em] \Rightarrow DC = \dfrac{36}{4} \\[1em] \Rightarrow DC = 9 \text{ cm.}

From figure,

AC = AD + DC = 4 + 9 = 13 cm.

Height BD = 6 cm.

Area of ΔABC = 12\dfrac{1}{2} × Base × Height

=12×AC×BD=12×13×6=13×3=39 cm2.= \dfrac{1}{2} \times AC \times BD \\[1em] = \dfrac{1}{2} \times 13 \times 6 \\[1em] = 13 \times 3 \\[1em] = 39 \text{ cm}^2.

The Assertion states Area is 40 cm2, which is incorrect.

So, Assertion (A) is false.

Areas of two similar triangles are proportional to the squares of their corresponding sides.

Reason (R) is true

A is false, R is true.

Hence, option 2 is the correct option.

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