Absolutely! Let's go through the step-by-step solution to determine the lengths of the legs of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, given its hypotenuse is [tex]\(10\sqrt{5}\)[/tex] inches.
### Step 1: Understand the Properties of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] Triangle
A [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is an isosceles right triangle, which means both of its legs are equal in length. Also, the relationship between the lengths of the legs [tex]\(a\)[/tex] and the hypotenuse [tex]\(c\)[/tex] is:
[tex]\[ c = a\sqrt{2} \][/tex]
In other words, the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as each leg.
### Step 2: Set Up the Relationship
Given that the hypotenuse [tex]\(c = 10\sqrt{5}\)[/tex] inches, we use the relationship:
[tex]\[ 10\sqrt{5} = a\sqrt{2} \][/tex]
### Step 3: Solve for [tex]\(a\)[/tex]
To find the leg length [tex]\(a\)[/tex], we need to isolate [tex]\(a\)[/tex] in the equation.
[tex]\[ a = \frac{10\sqrt{5}}{\sqrt{2}} \][/tex]
### Step 4: Simplify the Expression
Now, simplify the fraction:
[tex]\[ a = \frac{10\sqrt{5}}{\sqrt{2}} \][/tex]
The fraction can be rationalized, but we can directly calculate by recognizing that:
[tex]\[ \frac{\sqrt{5}}{\sqrt{2}} = \sqrt{\frac{5}{2}} \][/tex]
Thus,
[tex]\[ a = 10 \times \sqrt{\frac{5}{2}} \][/tex]
From here, the numerical calculations provide the solution as given in the results:
[tex]\[ a \approx 15.811388300841896 \][/tex]
### Final Results
Therefore, the lengths of the legs of the triangle are approximately [tex]\(15.811388300841896\)[/tex] inches each.
In summary, the hypotenuse is [tex]\(22.360679774997898\)[/tex] inches, and each leg is approximately [tex]\(15.811388300841896\)[/tex] inches.
