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The length of the longer leg of a right triangle is \( 22 \mathrm{ft} \mathrm{more} \mathrm{than} \mathrm{six} \mathrm{times} \mathrm{the} \mathrm{length} \mathrm{of} \mathrm{the} \mathrm{shorter} \mathrm{leg} \).The length of the hypotenuse is 23 ft more than six times the length of the shorter leg. Find the side lengths of the triangle. Length of the shorter leg: Length of the hypotenuse:

Ask by Herrera Lynch. in the United States
Jan 24,2025

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The shorter leg is 15 ft, the longer leg is 112 ft, and the hypotenuse is 113 ft.

Solución

Let's denote the length of the shorter leg as \( x \) and the length of the longer leg as \( y \). Given: 1. The length of the longer leg is \( 22 \) ft more than six times the length of the shorter leg: \[ y = 6x + 22 \] 2. The length of the hypotenuse is \( 23 \) ft more than six times the length of the shorter leg: \[ \text{Hypotenuse} = 6x + 23 \] We can use the Pythagorean theorem to find the side lengths of the triangle: \[ x^2 + y^2 = (\text{Hypotenuse})^2 \] Substitute the expressions for \( y \) and the hypotenuse into the Pythagorean theorem: \[ x^2 + (6x + 22)^2 = (6x + 23)^2 \] Now, we can solve this equation to find the value of \( x \), which will give us the length of the shorter leg. Once we have \( x \), we can find the length of the longer leg using the equation \( y = 6x + 22 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(x^{2}+\left(6x+22\right)^{2}=\left(6x+23\right)^{2}\) - step1: Expand the expression: \(37x^{2}+264x+484=36x^{2}+276x+529\) - step2: Move the expression to the left side: \(x^{2}-12x-45=0\) - step3: Factor the expression: \(\left(x-15\right)\left(x+3\right)=0\) - step4: Separate into possible cases: \(\begin{align}&x-15=0\\&x+3=0\end{align}\) - step5: Solve the equation: \(\begin{align}&x=15\\&x=-3\end{align}\) - step6: Rewrite: \(x_{1}=-3,x_{2}=15\) The solutions to the equation are \( x = -3 \) and \( x = 15 \). Since the length of a side cannot be negative, we discard the solution \( x = -3 \). Therefore, the length of the shorter leg is \( x = 15 \) ft. Now, we can find the length of the longer leg using the equation \( y = 6x + 22 \): \[ y = 6(15) + 22 = 90 + 22 = 112 \] So, the side lengths of the triangle are: - Length of the shorter leg: 15 ft - Length of the longer leg: 112 ft - Length of the hypotenuse: 6(15) + 23 = 90 + 23 = 113 ft

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Let's set the length of the shorter leg as \( x \). According to the problem, the longer leg can be expressed as \( 6x + 22 \), and the hypotenuse can be expressed as \( 6x + 23 \). Using the Pythagorean theorem, we can set up the equation: \[ (6x + 22)^2 + x^2 = (6x + 23)^2 \] Expanding both sides gives us: \[ (36x^2 + 264x + 484) + x^2 = (36x^2 + 276x + 529) \] Combining like terms leads to: \[ 37x^2 + 264x + 484 = 36x^2 + 276x + 529 \] Subtracting \(36x^2 + 276x + 529\) from both sides results in: \[ x^2 - 12x - 45 = 0 \] Factoring gives us: \[ (x - 15)(x + 3) = 0 \] Thus, \( x = 15 \) (since length cannot be negative). This means the length of the shorter leg is \( 15 \, \text{ft} \). Now we can find the other lengths: - The longer leg: \( 6(15) + 22 = 90 + 22 = 112 \, \text{ft} \) - The hypotenuse: \( 6(15) + 23 = 90 + 23 = 113 \, \text{ft} \) So the side lengths of the triangle are: Length of the shorter leg: 15 ft Length of the hypotenuse: 113 ft

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