Given Δ ABC ∼ Δ PQR.

Assertion (A): If area of Δ ABC : area of Δ PQR = 16 : 25, then perimeter of Δ ABC : perimeter of Δ PQR = 4 : 5.

Reason (R): The ratio of perimeter of two similar triangle is equal to the ratio of their corresponding sides.

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 ABC ∼ Δ PQR

If area of Δ ABC : area of Δ PQR = 16 : 25

We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\therefore \dfrac{\text{Area of Δ ABC}}{\text{Area of Δ PQR}} = \dfrac{AB^2}{PQ^2} \\[1em] \Rightarrow \dfrac{AB^2}{PQ^2} = \dfrac{16}{25} \\[1em] \Rightarrow \dfrac{AB}{PQ} = \sqrt{\dfrac{16}{25}} \\[1em] \Rightarrow \dfrac{AB}{PQ} = \dfrac{\sqrt{16}}{\sqrt{25}} \\[1em] \Rightarrow \dfrac{AB}{PQ} = \dfrac{4}{5}.$

Since, corresponding sides of similar triangle are proportional.

$\therefore \dfrac{AB}{PQ} = \dfrac{BC}{QR} = \dfrac{AC}{PR}$

We know that,

For any two or more equal ratios, each ratio is equal to the ratio between sum of their antecedents and sum of their consequents.

$\Rightarrow \dfrac{AB}{PQ} = \dfrac{AB + BC + AC}{PQ + QR + PR}\\[1em] \Rightarrow \dfrac{AB}{PQ} = \dfrac{\text{Perimeter of Δ ABC}}{\text{Perimeter of Δ PQR}} \\[1em] \Rightarrow \dfrac{4}{5} = \dfrac{\text{Perimeter of Δ ABC}}{\text{Perimeter of Δ PQR}} \\[1em]$

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.