To find the length of the hypotenuse in a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle where each leg measures 14 cm, we can use the properties of this special type of right triangle.
In a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle, the lengths of the legs are equal, and the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of one leg.
Given:
- Each leg of the triangle measures 14 cm.
Step-by-step solution:
1. Identify the relationship between the legs and the hypotenuse in a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle:
[tex]\[
\text{hypotenuse} = \text{leg length} \times \sqrt{2}
\][/tex]
2. Plug the given leg length into the formula:
[tex]\[
\text{hypotenuse} = 14 \, \text{cm} \times \sqrt{2}
\][/tex]
3. Multiply the leg length by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[
\text{hypotenuse} = 14 \sqrt{2} \, \text{cm}
\][/tex]
The answer is:
[tex]\[ 14 \sqrt{2} \, \text{cm} \][/tex]
Thus, the length of the hypotenuse in this [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle is [tex]\( 14 \sqrt{2} \)[/tex] cm. This corresponds to the choice [tex]\( 14 \sqrt{2} \, \text{cm} \)[/tex].
