Question

The shortest leg of a triangle is 1 feet shorter than the other leg.The hypotenuse of this triangle is 5 feet. what are the lengths of the two legs of this triangle?

The shortest leg is_____ ft long


The other leg is ________ft long

Answer 1

3 ft, 4 ft

Explanation:

Represent the length of the shortest leg using x. Then the other leg length is x + 1. Applying the Pythagorean Theorem,

x^2 + (x + 1)^2 = 5^2, or

x^2 + x^2 + 2x + 1 = 25, or

2x^2 + 2x - 24 = 0.

Reducing this, we get x^2 + x - 12 = 0, or (x + 4)(x - 3) =0.

Solving for x: x + 4 = 0 => x = -4; also x - 3 = 0 => x = 3

Since we're measuring length, let's omit x = -4 and focus on x = 3.

Shortest leg is 3 ft long; other leg length is 3 + 1 ft, or 4 ft, long.

Answer 2

• shortest: 3 ft
• other: 4 ft

Explanation:

The right triangle with hypotenuse 5 that has legs that differ by 1 is the 3-4-5 right triangle. The legs are 3 ft and 4 ft.

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Alternate solution

If you have never heard of a 3-4-5 right triangle, you can figure the leg lengths algebraically using the Pythagorean theorem. Let x represent the length of the "other" leg. Then the shortest is (x-1). The Pythagorean theorem tells you ...

5^2 = (x)^2 +(x -1)^2

25 = 2x^2 -2x +1

12 = x^2 -x = x(x -1)

Factors of 12 that differ by 1 are 3 and 4 or -3 and -4. We want x to be positive, so the value of interest is x=4

The "other" leg is 4 ft; the shortest leg is 3 ft.