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Difficulty: EasyPythagorean Theorem and Special Right Triangles

In right triangle ABCABC, the measure of angle BB is 90∘90^\circ and the measure of angle AA is 45∘45^\circ. If the length of leg ABAB is 88 inches, what is the length, in inches, of the hypotenuse ACAC?

  1. A
    88
  2. B
    424\sqrt{2}
  3. 828\sqrt{2}Answer
  4. D
    838\sqrt{3}
  5. E
    1616

Answer

The length of the hypotenuse is 828\sqrt{2} inches.
In right triangle ABCABC, the angle measures are 45∘45^\circ, 45∘45^\circ, and 90∘90^\circ. The lengths of the sides of a 45∘45^\circ-45∘45^\circ-90∘90^\circ triangle are in the ratio 1:1:21 : 1 : \sqrt{2}. Since the leg is 88 inches, the hypotenuse is 828\sqrt{2} inches.

Step-by-Step Solution

1
Identify the type of right triangle.
Since angle B=90∘B = 90^\circ and angle A=45∘A = 45^\circ, angle CC must also be 45∘45^\circ. This is a 45∘45^\circ-45∘45^\circ-90∘90^\circ special right triangle.
The sum of the angles in a triangle is always 180∘180^\circ.
2
Recall the ratio of the side lengths of a 45∘45^\circ-45∘45^\circ-90∘90^\circ triangle.
The ratio of the sides opposite the angles 45∘:45∘:90∘45^\circ : 45^\circ : 90^\circ is 1:1:21 : 1 : \sqrt{2}. Thus, the hypotenuse is equal to leg×2\text{leg} \times \sqrt{2}.
This is a standard geometric property of isosceles right triangles.
3
Calculate the length of the hypotenuse.
Multiply the leg length of 88 inches by 2\sqrt{2} to get 828\sqrt{2} inches.
The leg adjacent to the 45∘45^\circ angle is given as 88 inches.

Key Concept

Hypotenuse of a 45∘45^\circ-45∘45^\circ-90∘90^\circ special right triangle

Alternative Method

Alternatively, use the Pythagorean theorem: AB2+BC2=AC2AB^2 + BC^2 = AC^2. Since it is an isosceles right triangle, BC=AB=8BC = AB = 8. Thus, 82+82=AC2  ⟹  64+64=AC2  ⟹  AC=128=828^2 + 8^2 = AC^2 \implies 64 + 64 = AC^2 \implies AC = \sqrt{128} = 8\sqrt{2}.
Estimated Time:45s
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Topics

Pythagorean Theorem and Special Right Triangles

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