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  1. A carpenter is cutting a board to make a brace on a wall that will be used in
    a house. The shape of the brace will be a right triangle. In order for the brace
    to fit, the legs of the brace must be in a ratio of 3:2. The hypotenuse must be
    26 inches long. What will the lengths of the legs be? Round your answers to
    the nearest tenth of an inch.

Verified Answer :

GinnyAnswer

To determine the lengths of the legs of the right triangle brace where the legs are in a ratio of 3:2, and the hypotenuse is 26 inches long, follow these steps:

1. Understand the ratio and set up the problem:
- Let the lengths of the legs be \(\(3x\)\) and \(\(2x\)\), where \(\(x\)\) is a common multiplier.

2. Apply the Pythagorean theorem:
- The Pythagorean theorem states that in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse.
- This can be written as \(\((3x)^2 + (2x)^2 = 26^2\)\).

3. Substitute the values and simplify the equation:
- \(\( (3x)^2 + (2x)^2 = 26^2 \)\)
- Simplify the equation: \(\( 9x^2 + 4x^2 = 676 \)\)
- Combine like terms: \(\( 13x^2 = 676 \)\)

4. Solve for \(\(x\)\):
- \(\( x^2 = \frac{676}{13} \)\)
- \(\( x^2 = 52 \)\)
- Taking the square root of both sides: \(\( x = \sqrt{52} \)\)
- \(\( x \approx 7.2 \)\) (rounding to the nearest tenth)

5. Calculate the lengths of the legs using the ratio:
- \(\( \text{Leg 1} = 3x = 3 \times 7.2 = 21.6 \)\)
- \(\( \text{Leg 2} = 2x = 2 \times 7.2 = 14.4 \)\)

6. Round the results to the nearest tenth of an inch:
- \(\( \text{Leg 1} \approx 21.6 \)\)
- \(\( \text{Leg 2} \approx 14.4 \)\)

Therefore, the lengths of the legs are approximately \(\( 21.6 \)\) inches and \(\( 14.4 \)\) inches, while the hypotenuse is 26 inches.

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