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Question

Difficulty: EasyPythagorean Theorem and Special Right Triangles

In right triangle ABCABC, the hypotenuse ACAC has a length of 13 centimeters, and leg ABAB has a length of 5 centimeters. What is the length, in centimeters, of leg BCBC?

Answer: 12 centimeters

Answer

The length of leg BCBC is 12 centimeters.
Applying the Pythagorean theorem, we have 52+BC2=1325^2 + BC^2 = 13^2, which simplifies to 25+BC2=16925 + BC^2 = 169. Subtracting 25 from both sides gives BC2=144BC^2 = 144, and taking the square root of both sides gives BC=12BC = 12 centimeters.

Step-by-Step Solution

1
Set up the Pythagorean Theorem equation for right triangle ABCABC.
AB2+BC2=AC2AB^2 + BC^2 = AC^2
The Pythagorean Theorem states that in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse.
2
Substitute the given values for ABAB and ACAC into the formula.
52+BC2=1325^2 + BC^2 = 13^2
The length of leg ABAB is given as 5 centimeters, and the length of the hypotenuse ACAC is given as 13 centimeters.
3
Solve for the unknown leg length BCBC.
BC=12BC = 12
Squaring the values gives 25+BC2=16925 + BC^2 = 169. Subtracting 25 from both sides yields BC2=144BC^2 = 144. Taking the square root of both sides gives BC=12BC = 12.

Key Concept

Pythagorean Theorem

Alternative Method

Recognize the triangle as a standard 5-12-13 Pythagorean triple, which immediately gives the missing leg length of 12 without needing calculations.
Estimated Time:30s
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Topics

Pythagorean Theorem and Special Right Triangles

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