In a 45-45-90 triangle, the legs are equal and the lengths of the legs can be represented as \( x \). The length of the leg is given as 5 cm.
To find the length of the hypotenuse \( c \) in a 45-45-90 triangle, we use the formula:
\[ c = x\sqrt{2} \]
Substituting the length of the leg:
\[ c = 5\sqrt{2} \]
Now we can calculate \( 5\sqrt{2} \):
\[ 5\sqrt{2} \approx 5 \times 1.4142 \approx 7.071 \]
Next, we can find the perimeter \( P \) of the triangle, which is the sum of all sides:
\[ P = \text{leg} + \text{leg} + \text{hypotenuse} = 5 + 5 + 5\sqrt{2} \]
Substituting the value we calculated for the hypotenuse:
\[ P = 5 + 5 + 7.071 \approx 10 + 7.071 = 17.071 \]
Rounding to the nearest tenth:
\[ P \approx 17.1 \text{ cm} \]
Therefore, the perimeter of the triangle is 17.1 cm.