- Identify the triangle as an isosceles right triangle with the relationship \(h = l\sqrt{2}\) between the hypotenuse \(h\) and legs \(l\).
- Substitute the given value \(h = 10\sqrt{5}\) into the equation and solve for \(l = \frac{10\sqrt{5}}{\sqrt{2}}\).
- Simplify the expression by rationalizing the denominator to get \(l = 5\sqrt{10}\).
- The final leg length is \(\boxed{5\sqrt{10}}\).
Explanation
- Analyze the properties of a 45-45-90 triangle and set up the equation.
A \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle is an isosceles right triangle. Let \(l\) be the length of the legs and \(h\) be the length of the hypotenuse. According to the Pythagorean theorem, \(l^2 + l^2 = h^2\), which simplifies to \(2l^2 = h^2\), or \(h = l\sqrt{2}\). Given that the hypotenuse \(h\) measures \(10\sqrt{5}\) inches, we can set up the equation \(l\sqrt{2} = 10\sqrt{5}\).
- Solve for the leg length \(l\).
To find the length of one leg \(l\), we isolate \(l\) in the equation \(l\sqrt{2} = 10\sqrt{5}\). Dividing both sides by \(\sqrt{2}\) gives:
\[l = \frac{10\sqrt{5}}{\sqrt{2}}\]
We can simplify this by combining the square roots:
\[l = 10 \sqrt{\frac{5}{2}}\]
To rationalize the denominator, we multiply the numerator and denominator by \(\sqrt{2}\):
\[l = 10 \frac{\sqrt{5} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = 10 \frac{\sqrt{10}}{2} = 5\sqrt{10}\]
Thus, the length of one leg is \(5\sqrt{10}\) inches.
- Finalize the result.
Comparing our calculated result \(5\sqrt{10}\) with the given options:
A\(5\sqrt{5}\)
B\(10\sqrt{10}\)
C\(5\sqrt{10}\)
D\(10\sqrt{5}\)
The result matches option c. Therefore, the length of one leg is \(5\sqrt{10}\) inches. The final answer is \(\boxed{5\sqrt{10}}\).
Examples
In architectural design, when creating a triangular support bracket with a 45° angle, knowing the hypotenuse allows you to calculate the exact length of the required structural legs using the properties of a 45-45-90 triangle. This ensures that the support is geometrically precise and capable of bearing the intended load, maintaining the structural integrity and aesthetic balance of the building. For instance, if a support beam (the hypotenuse) measures \(10\sqrt{5}\) inches, calculating the legs as \(5\sqrt{10}\) inches provides the necessary measurements for cutting the materials accurately.