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 [tex]\(3x\)[/tex] and [tex]\(2x\)[/tex], where [tex]\(x\)[/tex] 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 [tex]\((3x)^2 + (2x)^2 = 26^2\)[/tex].
3. Substitute the values and simplify the equation:
- [tex]\( (3x)^2 + (2x)^2 = 26^2 \)[/tex]
- Simplify the equation: [tex]\( 9x^2 + 4x^2 = 676 \)[/tex]
- Combine like terms: [tex]\( 13x^2 = 676 \)[/tex]
4. Solve for [tex]\(x\)[/tex]:
- [tex]\( x^2 = \frac{676}{13} \)[/tex]
- [tex]\( x^2 = 52 \)[/tex]
- Taking the square root of both sides: [tex]\( x = \sqrt{52} \)[/tex]
- [tex]\( x \approx 7.2 \)[/tex] (rounding to the nearest tenth)
5. Calculate the lengths of the legs using the ratio:
- [tex]\( \text{Leg 1} = 3x = 3 \times 7.2 = 21.6 \)[/tex]
- [tex]\( \text{Leg 2} = 2x = 2 \times 7.2 = 14.4 \)[/tex]
6. Round the results to the nearest tenth of an inch:
- [tex]\( \text{Leg 1} \approx 21.6 \)[/tex]
- [tex]\( \text{Leg 2} \approx 14.4 \)[/tex]
Therefore, the lengths of the legs are approximately [tex]\( 21.6 \)[/tex] inches and [tex]\( 14.4 \)[/tex] inches, while the hypotenuse is 26 inches.
