To find the length of one leg in a \(45^\circ-45^\circ-90^\circ\) triangle given its hypotenuse, we need to use the properties of this special type of triangle.
1. In a \(45^\circ-45^\circ-90^\circ\) triangle, the legs are congruent (i.e., both legs have the same length).
2. The hypotenuse is \(\sqrt{2}\) times the length of each leg.
Given:
- Hypotenuse = 128 cm
Let's denote the length of each leg as \(x\).
According to the properties of a \(45^\circ-45^\circ-90^\circ\) triangle:
[tex]\[ \text{Hypotenuse} = x\sqrt{2} \][/tex]
Rearranging for \(x\):
[tex]\[ x = \frac{\text{Hypotenuse}}{\sqrt{2}} \][/tex]
Substituting the given value of the hypotenuse:
[tex]\[ x = \frac{128}{\sqrt{2}} \][/tex]
After performing the calculation:
[tex]\[ x \approx 90.51 \text{ cm} \][/tex]
Therefore, the length of one leg of the triangle is approximately \(90.51 \text{ cm}\).
The possible options given were:
- \(64 \text{ cm}\)
- \(64\sqrt{2} \text{ cm}\)
- \(128 \text{ cm}\)
- \(128\sqrt{2} \text{ cm}\)
The length of one leg (approximately \(90.51 \text{ cm}\)) corresponds to the option:
[tex]\[ 64\sqrt{2} \text{ cm} \approx 90.51 \text{ cm} \][/tex]
So, the correct answer is:
[tex]\[ 64\sqrt{2} \text{ cm} \][/tex]
