A steel plate is in the shape of a right triangle. The lengths of the legs are in the ratio 5:12. If the hypotenuse is 13 m, what is the area?

Correct — A.
Let the legs be 5k and 12k.

Pythagoras: (5k)² + (12k)² = 25k² + 144k² = 169k²
Hypotenuse = √(169k²) = 13k
Given 13k = 13, so k = 1

Legs = 5 m and 12 m
Area = ½ × 5 × 12 = 30 m² → option (a)

Why the others are wrong

• (b)
  60 m² — 60 m² is 5 × 12, the full product of the legs. A right triangle is half of the 5 × 12 rectangle, so the ½ has been dropped.
• (c)
  78 m² — 78 m² is ½ × 12 × 13, which uses the hypotenuse as the height. The hypotenuse is not perpendicular to either leg, so it cannot pair with one as base and height.
• (d)
  50 m² — 50 m² would need the legs to multiply to 100. The legs are 5 m and 12 m, whose product is 60, so the area is ½ × 60 = 30 m².

Area of a right triangle = ½ × leg × leg. The two legs meet at the right angle, so one is the base and the other the height.

When the legs come as a ratio, write them as 5k and 12k, express the hypotenuse in k with Pythagoras, then match it to the given length.

5, 12, 13 is a Pythagorean triple: 5² + 12² = 25 + 144 = 169 = 13². Spotting it gives k = 1 at once.

Key facts

• Area of a right triangle = ½ × the product of its two legs
• 5² + 12² = 13², so 5, 12, 13 is a Pythagorean triple
• Scaling every side by k scales the area by k²
• Other triples worth knowing: (3, 4, 5), (8, 15, 17), (7, 24, 25)

Common traps

• Forgetting the ½ and answering 5 × 12 = 60
• Using the hypotenuse as the height, which gives ½ × 12 × 13 = 78

No directly related past PYQ was found.