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Difficulty: MediumRight Triangles and the Pythagorean Theorem

An isosceles trapezoid has a shorter base of length 1010, a longer base of length 2222, and a height of 88. What is the perimeter of the trapezoid?

Answer: 52

Answer

The perimeter of the trapezoid is 5252.
To find the perimeter of the isosceles trapezoid, we first determine the length of the two congruent slanted sides. By drawing altitudes from the endpoints of the shorter top base to the longer bottom base, we create two right triangles at the sides of the trapezoid. The vertical leg of each right triangle is equal to the height of the trapezoid (88). The horizontal leg of each right triangle is equal to half the difference between the two base lengths, which is 22−102=6\frac{22 - 10}{2} = 6. Applying the Pythagorean theorem, the length of each slanted side is 62+82=10\sqrt{6^2 + 8^2} = 10. Finally, the perimeter is the sum of all four sides: 10+22+10+10=5210 + 22 + 10 + 10 = 52.

Step-by-Step Solution

1
Calculate the horizontal base of the right triangles formed by drawing heights from the top base to the bottom base.
66
An isosceles trapezoid is symmetrical. Drawing vertical lines representing the height of 88 from the endpoints of the shorter base of length 1010 down to the longer base of length 2222 divides the longer base into a middle segment of length 1010 and two equal end segments. The length of each end segment is 22−102=6\frac{22 - 10}{2} = 6.
2
Use the Pythagorean theorem to find the length of the slanted legs of the trapezoid.
1010
Each slanted leg is the hypotenuse of a right triangle with legs of length 66 and 88. According to the Pythagorean theorem, the hypotenuse length is 62+82=36+64=100=10\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10.
3
Find the perimeter of the trapezoid by adding the lengths of all four sides.
5252
The perimeter is the sum of the shorter base (1010), the longer base (2222), and the two slanted legs (each of length 1010). Therefore, the perimeter is 10+22+10+10=5210 + 22 + 10 + 10 = 52.

Key Concept

Applying the Pythagorean theorem to find missing side lengths in composite geometric figures
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Topics

Right Triangles and the Pythagorean Theorem

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