Question

for the right triangles below, find the exact values of the side lengths c and d.
if necessary, write your responses in simplified radical form.
c =

d =

Explanation:

Step1: Find the value of \( c \)

For the \( 45^{\circ}-45^{\circ}-90^{\circ} \) triangle, the legs are equal and the hypotenuse \( h = c\sqrt{2}\). Given \( h = 3 \), then \( c\sqrt{2}=3 \), so \( c=\frac{3}{\sqrt{2}}=\frac{3\sqrt{2}}{2} \).

Step2: Find the value of \( d \)

For the \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle, the hypotenuse \( d = 2\times \) shorter leg. The shorter leg is \( 6 \), so \( d = 2\times6 = 12 \).

Answer:

\( c=\frac{3\sqrt{2}}{2} \), \( d = 12 \)