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:
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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
