To solve for the length of the hypotenuse of a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle where each leg measures 12 cm, we need to use the properties of this special right-angled triangle.
For a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle, the legs are of equal length and the hypotenuse [tex]\(c\)[/tex] is given by the formula:
[tex]\[ c = \text{leg length} \times \sqrt{2} \][/tex]
Given that each leg of the triangle measures 12 cm, we can now find the hypotenuse length by substituting the leg length into the formula:
[tex]\[ c = 12 \, \text{cm} \times \sqrt{2} \][/tex]
Evaluating this expression, we get:
[tex]\[ c \approx 12 \, \text{cm} \times 1.414213562 \][/tex]
[tex]\[ c \approx 16.970562748477143 \, \text{cm} \][/tex]
Therefore, the length of the hypotenuse for the given [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle with each leg measuring 12 cm is approximately 16.97 cm. The correct choice is:
[tex]\[ 12 \sqrt{2} \, \text{cm} \][/tex]
