A right triangle has sides 3, 4, and 5. A smaller triangle is drawn inside it with its vertices on the sides of the larger triangle, such that it is similar to the larger triangle. If its perimeter is 6, what is its area?

Correct option is A

​Given :

A right triangle with sides 3, 4, 5
A smaller triangle is drawn inside it, similar to the larger triangle.
Perimeter of smaller triangle = 6.

Formula Used :

Perimeter ratio = ratio of corresponding sides
​$$\frac{P_{\text{small}}}{P_{\text{large}}} = k$$​​

Area ratio of similar triangles
​$$\frac{A_{\text{small}}}{A_{\text{large}}} = k^2$$​​

Area of right triangle
​$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$​​
Solution :

Perimeter of larger triangle
3 + 4 + 5 = 12

Scale factor
k$$= \frac{6}{12} = \frac{1}{2}$$​​

Area of larger triangle
​$$A_{\text{large}} = \frac{1}{2} \times 3 \times 4 = 6$$​​

Area of smaller triangle
​$$A_{\text{small}} = k^2 \times A_{\text{large}} = \left(\frac{1}{2}\right)^2 \times 6 = \frac{1}{4} \times 6 = 1.5$$​​