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Chapter 11: Problem 54

Write an equation and solve. The hypotenuse of a right triangle is \(\sqrt{34}\) in. long. The length of one leg is 1 in. less than twice the other leg. Find the lengths of the legs.

Short Answer

Expert verified
The lengths of the legs are 3 inches and 5 inches.

Step by step solution

01

Write the Pythagorean theorem

According to the Pythagorean theorem, for a right triangle with legs of length \(a\) and \(b\), and a hypotenuse of length \(c\), the following equation is true: \[a^2 + b^2 = c^2\]
02

Set up the equation using given information

We are given that \(c = \sqrt{34}\) and one leg is 1 inch less than twice the other leg. Let鈥檚 call one leg \(a\) and the other leg \((2a - 1)\). We can write the equation from the Pythagorean theorem as: \[a^2 + (2a - 1)^2 = (\sqrt{34})^2\]
03

Solve the equation

First, simplify the equation: \[a^2 + (4a^2 - 4a + 1) = 34\] Now, combine the like terms: \[5a^2 - 4a + 1 = 34\] To solve the quadratic equation, first subtract 34 from both sides to set it to zero: \[5a^2 - 4a - 33 = 0\] Now, factor the quadratic equation. You can check that the correct factorization is: \[(a - 3)(5a + 11) = 0\]
04

Find the possible values for the leg length

Set each factor to zero and solve for \(a\): 1. \(a - 3 = 0\) \(a = 3\) 2. \(5a + 11 = 0\) \(5a = -11\) \(a = -\dfrac{11}{5}\) (discard this value as length cannot be negative) Since \(a\) must be positive, we have \(a = 3\). Now, plug this value into the expression for the other leg length: \[b = 2a - 1 = (2 \cdot 3) - 1 = 5\]
05

State the lengths of the legs

The lengths of the legs are 3 inches and 5 inches.

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