Question

Assertion (A): If $$\triangle ABC\sim\triangle DEF$$, $$\operatorname{ar}(\triangle ABC) = 81\ \text{cm}^{2}$$, $$\operatorname{ar}(\triangle DEF) = 144\ \text{cm}^{2}$$ and $$EF = 24\ \text{cm}$$ then $$BC = 18\ \text{cm}$$.

Reason (R): $$\frac{\operatorname{ar}(\triangle ABC)}{\operatorname{ar}(\triangle DEF)}=\frac{AB^{2}}{DE^{2}}=\frac{BC^{2}}{EF^{2}}=\frac{AC^{2}}{DF^{2}}$$.

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. Using the area ratio and the corresponding sides $$BC$$ and $$EF$$: $$\frac{81}{144}=\frac{BC^{2}}{24^{2}}$$

$$BC=24\sqrt{\frac{81}{144}}=24\times\frac{3}{4}=18\ \text{cm}$$

Step 2 – Reason: R is true. For similar triangles, the ratio of their areas equals the square of the ratio of corresponding sides.

Step 3 – Link: Applying the area-to-side-square ratio to $$BC$$ and $$EF$$ gives the asserted length.