The long leg of a right triangle is 97 feet longer than the short leg. The hypotenuse is 113 feet long. How long are the legs of the right triangle?

Denote the length of the short leg of the right triangle as "x" feet. According to the problem, the long leg is 97 feet longer than the short leg, so the length of the long leg is "x + 97" feet.

The hypotenuse of the right triangle is 113 feet long.

Now, use the Pythagorean theorem to relate the lengths of the legs and the hypotenuse:

c² = a² + b²

Here, c represents the length of the hypotenuse, and a and b represent the lengths of the legs. In our case:

113² = x² + (x + 97)²

Now, solve for x:

113² = x² + (x + 97)²

113² = x² + (x² + 194x + 97²)

113² = x² + x² + 194x + 97²

113² = 2x² + 194x + 97²

Now, simplify the equation:

2x² + 194x + 97² - 113² = 0

Next, use the quadratic formula to solve for x:

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 2, b = 194, and c = 97² - 113². Plugging in these values:

x = (-194 ± √(194² - 4(2)(97² - 113²))) / (2(2))

x = (-194 ± √(37636 - 4(2)(-322))) / 4

x = (-194 ± √(37636 + 2576)) / 4

x = (-194 ± √(40212)) / 4

Now, calculate the two possible values of x:

x₁ = (-194 + √40212) / 4

x₂ = (-194 - √40212) / 4

x₁ ≈ 31.76

x₂ ≈ -60.76

Since the length of a leg cannot be negative, discard the negative solution.

So, the short leg of the right triangle is approximately 31.76 feet, and the long leg is x + 97, which is approximately:

31.76 + 97 = 128.76 feet.

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sqdancefan sqdancefan · Answer 2 · 2024-09-25T02:13:22-04:00

Answer:

15 ft, 112 ft

Step-by-step explanation:

You want the lengths of the legs of a right triangle with a hypotenuse of 113 ft if the longer leg is 97 ft longer than the shorter leg.

Pythagorean theorem

We can use the Pythagorean theorem to relate the lengths of the legs and the hypotenuse. If x is the short leg, then x+97 is the long leg. The relationship is ...

x² + (x +97)² = 113²

2x² +194x +9409 = 12769 . . . . . simplify

x² +97x -1680 = 0 . . . . . . . . . . subtract 12769

(x -15)(x +112) = 0 . . . . . . . . . factor

x = 15 . . . . . . . . . . . . . positive solution

The short leg is 97 feet; the long leg is 112 feet.

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