To find the length of the hypotenuse in a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle where each leg measures 12 cm, we can utilize the properties of this specific type of triangle.
A [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is an isosceles right triangle, meaning that its legs are of equal length. The ratio of the lengths of the sides in such a triangle is:
[tex]\[
1 : 1 : \sqrt{2}
\][/tex]
If each leg (the two shorter sides) has a length of 12 cm, then we set one leg to 12 cm.
For a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the hypotenuse ([tex]\(c\)[/tex]) can be found using the relationship:
[tex]\[
c = \text{leg} \times \sqrt{2}
\][/tex]
Given that each leg is 12 cm long, we substitute this value into the equation:
[tex]\[
c = 12 \times \sqrt{2}
\][/tex]
Therefore, the length of the hypotenuse can be expressed as:
[tex]\[
12\sqrt{2} \text{ cm}
\][/tex]
Among the choices provided:
- 6 cm
- [tex]\(6 \sqrt{2} \text{ cm}\)[/tex]
- 12 cm
- [tex]\(12 \sqrt{2} \text{ cm}\)[/tex]
The correct answer is:
[tex]\[
12 \sqrt{2} \text{ cm}
\][/tex]
