The hypotenuse of a triangle is 52 ft long. The length of
The problem involves a right triangle where the hypotenuse measures 52 feet, and one of the legs is longer than the other by 28 feet. The goal is to find the lengths of both legs.
Let the length of the shorter leg be denoted as x feet. Then, the longer leg will be x + 28 feet. According to the Pythagorean theorem, the sum of the squares of the legs equals the square of the hypotenuse:
Feel free to substitute and solve the resulting quadratic equation to determine the exact lengths of the legs.
Paper For Above instruction
The problem presented is a classic application of the Pythagorean theorem in geometry, which states that for a right triangle, the sum of the squares of the lengths of the two legs (the sides adjacent to the right angle) equals the square of the hypotenuse. Specifically, given a hypotenuse of length 52 feet and a relationship between the legs, this problem encompasses several key principles of geometric problem-solving and algebraic manipulation.
Let the length of the shorter leg be x feet. Since the other leg is longer by 28 feet, its length is x + 28 feet. Applying the Pythagorean theorem, the equation can be written as:
x² + (x + 28)² = 52²
Expanding the equation yields:
x² + x² + 56x + 784 = 2704
which simplifies to:
2x² + 56x + 784 = 2704
Subtract 2704 from both sides:
2x² + 56x + 784 - 2704 = 0 resulting in:
2x² + 56x - 1920 = 0
Dividing through by 2 to simplify:
x² + 28x - 960 = 0
This quadratic equation can be solved using the quadratic formula:
x = [-b ± √(b² - 4ac)] / 2a
where a = 1, b = 28, and c = -960. Calculating the discriminant:
∆ = 28² - 4(1)(-960) = 784 + 3840 = 4624
Taking the square root:
√4624 ≈ 67.98
Now, solving for x:
x = [-28 ± 67.98] / 2
Thus, two solutions are obtained:
x ≈ (-28 + 67.98) / 2 ≈ 39.98 / 2 ≈ 19.99 ft and
x ≈ (-28 - 67.98) / 2 ≈ -95.98 / 2 ≈ -47.99 ft
Since a length cannot be negative, the valid solution for the shorter leg is approximately 20 feet. Consequently, the longer leg measures approximately 20 + 28 = 48 feet.
In conclusion, the shorter leg is approximately 20 feet long, and the longer leg is approximately 48 feet long. These solutions satisfy the original Pythagorean relation, ensuring the calculation's accuracy and relevancy to the geometry problem presented.
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