A right-angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. A second right-angled triangle is similar to the first, and its hypotenuse is 39 cm. What is the area of the second triangle?
270 cm²
Step 1 – find the other leg of the first triangle (Pythagoras):
\(\text{other leg} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ cm}\)
Step 2 – area of the first triangle:
\(A_1 = \dfrac{1}{2} \times 5 \times 12 = 30 \text{ cm}^2\)
Step 3 – scale factor between the two similar triangles:
Hypotenuse ratio = \(\dfrac{39}{13} = 3\), so the linear scale factor is 3.
Step 4 – areas scale by the square of the linear scale:
\(A_2 = A_1 \times 3^2 = 30 \times 9 = 270 \text{ cm}^2\)
Hence the area of the second triangle is 270 cm².
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