Question

Assertion (A): In the given figure, $$MN\parallel BC$$ and $$AM : MB = 1 : 2$$. Then, $$\frac{\operatorname{ar}(\triangle AMN)}{\operatorname{ar}(\triangle ABC)}=\frac{1}{4}$$.

Reason (R): The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

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

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

Explanation:

Step 1 – Assertion: A is false. Since $$AM:MB=1:2$$, we have $$AM:AB=1:3$$. The triangles are similar, so their area ratio is the square of the corresponding side ratio: $$\frac{\operatorname{ar}(\triangle AMN)}{\operatorname{ar}(\triangle ABC)}=\left(\frac{AM}{AB}\right)^{2}=\left(\frac{1}{3}\right)^{2}=\frac{1}{9}$$

Step 2 – Reason: R is true. The area ratio of similar triangles equals the square of the ratio of corresponding sides.