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Question

Difficulty: EasyTriangles: Properties, Perimeter, and Area

In a right triangle, the lengths of the two legs are in a ratio of 3:43:4. If the perimeter of the triangle is 3636 centimeters, what is the area of the triangle, in square centimeters?

Answer: 54 square centimeters

Answer

The area of the triangle is 54 square centimeters.
Since the ratio of the legs of the right triangle is 3:4, the triangle forms a classic 3-4-5 right triangle proportion. The perimeter is 3x+4x+5x=12x3x + 4x + 5x = 12x. Setting 12x=3612x = 36 yields x=3x = 3. The legs are therefore 99 cm and 1212 cm. Calculating the area using 12×9×12\frac{1}{2} \times 9 \times 12 gives 5454 square centimeters.

Step-by-Step Solution

1
Express the side lengths in terms of a variable xx
Legs are 3x3x and 4x4x, and hypotenuse is 5x5x
By the Pythagorean theorem, a right triangle with legs in ratio 3:4 has hypotenuse ratio 32+42=5\sqrt{3^2 + 4^2} = 5.
2
Solve for xx using the given perimeter
x=3x = 3
The sum of all three sides is 3x+4x+5x=12x=363x + 4x + 5x = 12x = 36, giving x=3x = 3.
3
Calculate the actual leg lengths and area
Legs are 99 cm and 1212 cm; Area is 5454 cm2\text{cm}^2
The area of a right triangle is half the product of its perpendicular legs: 12×9×12=54\frac{1}{2} \times 9 \times 12 = 54.

Key Concept

Perimeter and area of right triangles using standard side ratios
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Topics

Triangles: Properties, Perimeter, and Area

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