Question

Assertion (A): In $$\triangle ABC$$, $$P$$ and $$Q$$ are points on $$AB$$ and $$AC$$ respectively such that $$AP = 3\ \text{cm}$$, $$PB = 6\ \text{cm}$$, $$AQ = 5\ \text{cm}$$ and $$QC = 10\ \text{cm}$$. Then, $$BC = 3PQ$$.

Reason (R): The corresponding sides of similar triangles $$APQ$$ and $$ABC$$ are proportional.

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. We have $$AB=3+6=9\ \text{cm}$$ and $$AC=5+10=15\ \text{cm}$$, so $$\frac{AP}{AB}=\frac{AQ}{AC}=\frac{1}{3}$$. The triangles are similar, giving $$\frac{PQ}{BC}=\frac{1}{3}$$ or $$BC=3PQ$$.

Step 2 – Reason: R is true. Corresponding sides of similar triangles are proportional.

Step 3 – Link: The proportional sides of $$\triangle APQ$$ and $$\triangle ABC$$ give the asserted relationship between $$BC$$ and $$PQ$$.