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?

Correct — D.
First triangle, missing leg: √(13² − 5²) = √144 = 12 cm
Its area: ½ × 5 × 12 = 30 cm²

Scale factor: 39 ÷ 13 = 3
Areas scale by its square: 3² = 9
Second area: 30 × 9 = 270 cm² → option (d)

Why the others are wrong

• (a)
  30 cm² — 30 cm² is the first triangle's area, before any scaling. The second triangle has a 39 cm hypotenuse, three times as long, so its area must be larger.
• (b)
  90 cm² — 90 cm² scales the area by 3, the length ratio. Area is length × length, so it scales by 3² = 9: 30 × 9 = 270, not 30 × 3.
• (c)
  160 cm² — 160 cm² is not 30 × 3². Check directly: the second triangle's legs are 5 × 3 = 15 and 12 × 3 = 36, and ½ × 15 × 36 = 270.

In similar figures, every length is multiplied by the same scale factor k. An area is a length times a length, so it is multiplied by k², and a volume by k³.

Here k = 39 ⁄ 13 = 3, so the area grows ninefold. Scaling the sides first gives the same answer: 5, 12, 13 become 15, 36, 39, and ½ × 15 × 36 = 270.

Key facts

• Ratio of areas of similar triangles = (ratio of corresponding sides)².
• 5, 12, 13 is a Pythagorean triple: 5² + 12² = 13².
• Perimeters, heights and medians of similar triangles scale by k, like the sides.

Common traps

• Multiplying the area by the length ratio 3 instead of its square 9
• Stopping at the first triangle's area, 30 cm², which is printed among the options

No directly related past PYQ was found.