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The hypotenuse of a right triangle is 50 inches long. The triangle's longer leg is 5 inches longer than its shorter leg. Which equation can you use to find the length of the triangle's shorter leg, x ? (1)/(2)x(x+5)=50 x^(2)+(x+5)^(2)=50^(2) To the nearest tenth of an inch, how long is the triangle's shorter leg? □ inches

The hypotenuse of a right triangle is 5050 inches long. The triangle's longer leg is 55 inches longer than its shorter leg.\newlineWhich equation can you use to find the length of the triangle's shorter leg, xx?\newline12x(x+5)=50\frac{1}{2}x(x+5)=50\newlinex2+(x+5)2=502x^{2}+(x+5)^{2}=50^{2}\newlineTo the nearest tenth of an inch, how long is the triangle's shorter leg?\newline□\square inches

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Q. The hypotenuse of a right triangle is 5050 inches long. The triangle's longer leg is 55 inches longer than its shorter leg.\newlineWhich equation can you use to find the length of the triangle's shorter leg, xx?\newline12x(x+5)=50\frac{1}{2}x(x+5)=50\newlinex2+(x+5)2=502x^{2}+(x+5)^{2}=50^{2}\newlineTo the nearest tenth of an inch, how long is the triangle's shorter leg?\newline□\square inches
  1. Identify Equation: Identify the correct equation to represent the relationship between the sides of the right triangle.\newlineThe Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse cc is equal to the sum of the squares of the lengths of the other two sides aa and bb. The equation is a2+b2=c2a^2 + b^2 = c^2.\newlineSince the hypotenuse is 5050 inches and the longer leg is 55 inches longer than the shorter leg, we can represent the shorter leg as xx and the longer leg as x+5x + 5.\newlineThe correct equation to represent this relationship is x2+(x+5)2=502x^2 + (x + 5)^2 = 50^2.
  2. Solve Equation: Solve the equation for xx. First, expand the equation: x2+(x+5)2=502x^2 + (x + 5)^2 = 50^2 becomes x2+x2+10x+25=2500x^2 + x^2 + 10x + 25 = 2500. Combine like terms: 2x2+10x+25=25002x^2 + 10x + 25 = 2500. Subtract 25002500 from both sides to set the equation to zero: 2x2+10x+25−2500=02x^2 + 10x + 25 - 2500 = 0. Simplify the equation: 2x2+10x−2475=02x^2 + 10x - 2475 = 0. Divide all terms by 22 to simplify further: x2+5x−1237.5=0x^2 + 5x - 1237.5 = 0.
  3. Use Quadratic Formula: Use the quadratic formula to solve for xx. The quadratic formula is x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=1a = 1, b=5b = 5, and c=−1237.5c = -1237.5. Calculate the discriminant b2−4acb^2 - 4ac: (5)2−4(1)(−1237.5)=25+4950=4975(5)^2 - 4(1)(-1237.5) = 25 + 4950 = 4975. Calculate the square root of the discriminant: 4975≈70.53\sqrt{4975} \approx 70.53. Apply the quadratic formula: x=−5±70.532×1x = \frac{-5 \pm 70.53}{2 \times 1}.
  4. Find Solutions: Find the two possible solutions for xx.\newlineFirst solution: x=(-5+70.53)/2≈32.765x = (\text{-}5 + 70.53) / 2 \approx 32.765.\newlineSecond solution: x=(-5−70.53)/2≈-37.765x = (\text{-}5 - 70.53) / 2 \approx \text{-}37.765.\newlineSince a leg length cannot be negative, the second solution is not physically possible.\newlineTherefore, the length of the shorter leg is approximately 32.76532.765 inches.
  5. Round to Nearest Tenth: Round the length of the shorter leg to the nearest tenth of an inch.\newlineRounded to the nearest tenth, the length of the shorter leg is approximately 32.832.8 inches.

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