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Difficulty: EasyPythagorean Theorem and Special Right Triangles

A right triangle has two legs of equal length. If the hypotenuse of the triangle is 10210\sqrt{2} centimeters, what is the length, in centimeters, of one of the legs?

Answer: 10 centimeters

Answer

The length of one of the legs is 10 centimeters.
An isosceles right triangle possesses acute angles of 45∘45^\circ and side ratios of x:x:x2x : x : x\sqrt{2}, where xx represents the leg length. Given a hypotenuse of 10210\sqrt{2} centimeters, we equate x2=102x\sqrt{2} = 10\sqrt{2}. Dividing both sides of the equation by 2\sqrt{2} isolates the leg length, giving x=10x = 10 centimeters.

Step-by-Step Solution

1
Determine the triangle type from the given properties.
The triangle is a 45∘45^\circ-45∘45^\circ-90∘90^\circ special right triangle (isosceles right triangle).
A right triangle with two legs of equal length must have acute angles measuring 45∘45^\circ each, making it an isosceles right triangle.
2
Set up an equation utilizing the ratios of the side lengths.
Let xx be the leg length. The hypotenuse length is represented by x2=102x\sqrt{2} = 10\sqrt{2} centimeters.
The hypotenuse of a 45∘45^\circ-45∘45^\circ-90∘90^\circ special right triangle is always 2\sqrt{2} times the length of one of its legs.
3
Solve the equation for the variable xx.
x=10x = 10
Dividing both sides of the equation by 2\sqrt{2} isolates the variable xx representing the leg length.

Key Concept

Properties of 45∘45^\circ-45∘45^\circ-90∘90^\circ special right triangles.

Alternative Method

Alternatively, you can apply the Pythagorean Theorem: a2+b2=c2a^2 + b^2 = c^2. Since both legs are equal in length, we can set a=b=xa = b = x. This yields the equation x2+x2=(102)2x^2 + x^2 = (10\sqrt{2})^2. Simplifying both sides gives 2x2=100×2=2002x^2 = 100 \times 2 = 200. Dividing by 2 yields x2=100x^2 = 100, and taking the square root of both sides gives x=10x = 10 centimeters.
Estimated Time:45s
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Topics

Pythagorean Theorem and Special Right Triangles

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