The hypotenuse of a [tex]45^{\circ}-45^{\circ}-90^{\circ}[/tex] triangle measures 128 cm. What is the length of one leg of the triangle?

A. [tex]64 \, \text{cm}[/tex]
B. [tex]64 \sqrt{2} \, \text{cm}[/tex]
C. [tex]128 \, \text{cm}[/tex]
D. [tex]128 \sqrt{2} \, \text{cm}[/tex]



Answer :

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]