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Answer :
Final answer:
The Pythagorean theorem can be used to find the lengths of the other two sides of a right triangle. By substituting the given values into the equation, you can solve for a or b. Once you have the lengths of the other two sides, you can use the Pythagorean theorem to check your answer.
Explanation:
In a right triangle, the Pythagorean theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse.
To find the other two lengths, you can use the given information. The length of the hypotenuse (h) is 13.0, the length of one segment of the hypotenuse (ha) is 5.60, and the other leg length (lo) is 7.83.
You can solve the equation a² + b² = c² for a or b by substituting the given values. For example, substituting 5.60 for a and 7.83 for b, you get (5.60)² + (7.83)² = (13.0)². Solving this equation will give you the lengths of the other two sides to the nearest tenth.
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Final Answer:
The lengths of the legs of the right triangle are approximately l = 8.8 and la = 6.6.
Explanation:
Let's denote the length of the hypotenuse as h, the length of the shorter leg as l, and the length of the longer leg as la. Additionally, let ha and hi represent the lengths of the segments of the hypotenuse formed by the bisector of the right angle.
We are given that h = 13.0, ha = 5.60, and l0 = 7.83. First, we can find hi using the property that the lengths of the segments of the hypotenuse are proportional to the lengths of the legs. Therefore, hi/ha = l0/l, which implies hi = (ha * l0) / l.
Substituting the given values, hi = (5.60 * 7.83) / 13.0 ≈ 3.37.
Now, we can find the length of the shorter leg using the Pythagorean Theorem: l² + hi² = la². Substituting the values, we get l² + (3.37)² = la². Solving for l, we find l ≈ 8.8.
Finally, to find the length of the longer leg, we use the fact that la = h - hi. Substituting the values, la = 13.0 - 3.37 ≈ 9.7.
Therefore, the lengths of the legs of the right triangle are approximately l = 8.8 and la = 6.6.
