Given Δ PQR ∼ Δ DEF.

Assertion (A): If area of Δ PQR : area of Δ DEF = 9 : 49, then the ratio of their corresponding medians is also 4 : 9.

Reason (R): For the similar triangles, the ratio of their corresponding sides is equal to the ratio of their corresponding medians.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Given Δ PQR ∼ Δ DEF and PX is median of triangle PQR, DY is median of triangle DEF.

Since Δ PQR ∼ Δ DEF, corresponding sides of similar triangle are proportional.

$\therefore\dfrac{PQ}{DE} = \dfrac{QR}{EF}\\[1em] \Rightarrow \dfrac{PQ}{DE} = \dfrac{\dfrac{1}{2}QR}{\dfrac{1}{2}EF}\\[1em] \Rightarrow \dfrac{PQ}{DE} = \dfrac{QX}{EY}$

And we also know that corresponding angles of similar triangles are equal.

∴ ∠Q = ∠E

Now, in Δ PQX and Δ DEY,

$\Rightarrow \dfrac{PQ}{DE} = \dfrac{QX}{EY}$

⇒ ∠Q = ∠E

Using SAS similarity,

⇒ Δ PQX ∼ Δ DEY

Since, corresponding sides of similar triangle are proportional,

$\Rightarrow \dfrac{PQ}{DE} = \dfrac{QX}{EY} = \dfrac{PX}{DY}$

If two triangles are similar, then the ratio of their areas equals the square of the ratio of their corresponding sides.

$\therefore\dfrac{\text{area of ΔPQR}}{\text{area of ΔDEF}} = \dfrac{PQ^2}{DE^2} = \dfrac{PX^2}{DY^2}$

So, for the similar triangles, the ratio of their corresponding sides is equal to the ratio of their corresponding medians.

So, reason (R) is true.

Given,

area of Δ PQR : area of Δ DEF = 9 : 49

$\Rightarrow \dfrac{\text{area of ΔPQR}}{\text{area of ΔDEF}} = \dfrac{9}{49} \\[1em] \Rightarrow \dfrac{PX^2}{DY^2} = \dfrac{9}{49}\\[1em] \Rightarrow \dfrac{PX}{DY} = \dfrac{\sqrt{9}}{\sqrt{49}}\\[1em] \Rightarrow \dfrac{PX}{DY} = \dfrac{3}{7}$

So, assertion (A) is false.

Thus, Assertion (A) is false, but Reason (R) is true.

Hence, option 2 is the correct option.