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Difficulty: EasyTriangles and Pythagorean Theorem

A rectangular garden gate needs a diagonal wooden brace for support. The gate is 44 feet tall and 33 feet wide. What is the length, in feet, of the diagonal brace?

Answer: 5 feet

Answer

5
The diagonal brace forms the hypotenuse of a right triangle where the width (33 feet) and height (44 feet) are the legs. According to the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we substitute the given lengths: 32+42=c23^2 + 4^2 = c^2. This simplifies to 9+16=259 + 16 = 25. Taking the square root of 2525 gives a diagonal length of 55 feet. This forms a common 3−4−53-4-5 Pythagorean triple.

Step-by-Step Solution

1
Identify the right triangle formed by the gate
The legs of the right triangle are 33 and 44, and the hypotenuse is the unknown diagonal brace.
The corners of a rectangular gate form right angles, so drawing a diagonal creates a right triangle.
2
Apply the Pythagorean theorem
32+42=c23^2 + 4^2 = c^2
The Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) is used to find the missing side length of a right triangle.
3
Calculate the squares of the legs
9+16=c29 + 16 = c^2
Squaring the given lengths prepares the equation for addition.
4
Solve for the hypotenuse
25=c225 = c^2, which means c=5c = 5
Taking the square root of the sum yields the length of the diagonal brace.

Key Concept

Using the Pythagorean theorem to find the hypotenuse of a right triangle.
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Topics

Triangles and Pythagorean Theorem

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