Each leg of a [tex]45^{\circ}-45^{\circ}-90^{\circ}[/tex] triangle measures [tex]12 \, \text{cm}[/tex]. What is the length of the hypotenuse?

A. [tex]6 \, \text{cm}[/tex]
B. [tex]6 \sqrt{2} \, \text{cm}[/tex]
C. [tex]12 \, \text{cm}[/tex]
D. [tex]12 \sqrt{2} \, \text{cm}[/tex]



Answer :

Certainly! Let's solve the problem step by step.

We are dealing with a [tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex] triangle. In this type of triangle, the legs are of equal length, and the hypotenuse is \( \sqrt{2} \) times the length of each leg.

Given:
- Each leg of the triangle measures \( 12 \) cm.

To find the length of the hypotenuse, we use the relationship specific to a [tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex] triangle:
[tex]\[ \text{Hypotenuse} = \text{leg} \times \sqrt{2} \][/tex]

Substituting the given leg length:
[tex]\[ \text{Hypotenuse} = 12 \, \text{cm} \times \sqrt{2} \][/tex]

Now, calculate the result:
[tex]\[ \text{Hypotenuse} = 12 \sqrt{2} \, \text{cm} \][/tex]

Thus, the length of the hypotenuse is:
[tex]\[ 12 \sqrt{2} \, \text{cm} \][/tex]

So the correct choice from the given options is:
[tex]\[ \boxed{12 \sqrt{2} \, \text{cm}} \][/tex]