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Difficulty: EasyPythagorean Theorem and Special Right Triangles

A square tabletop has a diagonal length of 88 feet. What is the area of the tabletop, in square feet?

  1. A
    16
  2. B
    16216\sqrt{2}
  3. 32Answer
  4. D
    32232\sqrt{2}
  5. E
    64

Answer

32
The correct answer is 32. The diagonal of a square divides it into two congruent 45-45-90 right triangles, where the diagonal is the hypotenuse. The ratio of the diagonal to the side length in a 45-45-90 triangle is 2\sqrt{2} to 11. Therefore, a square with a diagonal of 88 feet has a side length of s=82s = \frac{8}{\sqrt{2}} feet. The area of the square is s2=(82)2=642=32s^2 = \left(\frac{8}{\sqrt{2}}\right)^2 = \frac{64}{2} = 32 square feet.

Step-by-Step Solution

1
Relate the diagonal of a square to its side length using special right triangles.
The diagonal of a square splits the square into two 45-45-90 right triangles. The hypotenuse of these triangles is the diagonal, 88 feet, and the legs are the sides of the square, ss. The relationship is s2=8s\sqrt{2} = 8.
In a 45-45-90 triangle, the hypotenuse is 2\sqrt{2} times the length of a leg.
2
Solve for the side length ss of the square tabletop.
s=82s = \frac{8}{\sqrt{2}} feet.
Divide both sides of the equation by 2\sqrt{2} to isolate the side length ss.
3
Calculate the area of the square tabletop.
Area =s2=(82)2=642=32= s^2 = \left(\frac{8}{\sqrt{2}}\right)^2 = \frac{64}{2} = 32 square feet.
The area of a square is calculated by squaring its side length.

Key Concept

Using the properties of 45-45-90 special right triangles to find side lengths and area from the diagonal of a square.
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Topics

Pythagorean Theorem and Special Right Triangles

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