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?
1.5
Step 1 – Find similarity ratio from perimeters.
Original perimeter = \(3 + 4 + 5 = 12\); new perimeter = \(6\).
\(k = \dfrac{6}{12} = \dfrac{1}{2}\)
Step 2 – Original area.
\(A_{\text{orig}} = \dfrac{1}{2} \times 3 \times 4 = 6\)
Step 3 – New area uses \(A_{\text{new}} = k^2 \cdot A_{\text{orig}}\):
\(A_{\text{new}} = \left(\dfrac{1}{2}\right)^2 \times 6 = \dfrac{6}{4} = 1.5\)
Hence the correct answer is option 1) 1.5.
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