Certainly! Let's break it down step by step. We need to find the length of the legs of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle where the hypotenuse measures 8 cm.
1. Understanding 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.
- The legs of the triangle are equal in length.
- The ratio of the sides of this triangle is [tex]\(1:1:\sqrt{2}\)[/tex].
2. Relating the Hypotenuse to the Legs:
- If we let the length of each leg be [tex]\( x \)[/tex], then the hypotenuse is [tex]\( x\sqrt{2} \)[/tex].
3. Given Hypotenuse and Finding the Legs:
- We are given that the hypotenuse measures 8 cm. Therefore, we set up the equation:
[tex]\[
x\sqrt{2} = 8
\][/tex]
4. Solving for [tex]\( x \)[/tex]:
- To find the length of each leg ([tex]\( x \)[/tex]), we divide both sides by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[
x = \frac{8}{\sqrt{2}}
\][/tex]
5. Simplification:
- We can simplify the fraction by multiplying the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[
x = \frac{8}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{8\sqrt{2}}{2} = 4\sqrt{2}
\][/tex]
6. Calculation of [tex]\( x \)[/tex] (Numerical Value):
- From our known answer, the approximate value for [tex]\( 4\sqrt{2} \)[/tex] is approximately [tex]\( 5.65685424949238 \)[/tex] cm.
Therefore, the length of each leg of the [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, when the hypotenuse measures 8 cm, is:
[tex]\[
4\sqrt{2} \text{ cm} \approx 5.65685424949238 \text{ cm}
\][/tex]
So, 5.65685424949238 cm would be one of the possible values for the length of each leg of this triangle. None of the given options (4 cm, 6 cm, 9 cm, [tex]\(9 \sqrt{2} \)[/tex] cm, 18 cm, [tex]\(18 \sqrt{2} \)[/tex] cm) directly fit this exact length.
Therefore, if the length of the legs needs to match one of the provided values, more context about the question might be necessary. Based on the standard properties of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the leg length is [tex]\(4\sqrt{2}\)[/tex] cm.
