Question

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

Explanation:

Step1: Find the value of \( b \)

In a \( 45^{\circ}-45^{\circ}-90^{\circ} \) triangle, the sides are in the ratio \( 1:1:\sqrt{2} \).
If the legs are of length \( x \), the hypotenuse is \( x\sqrt{2} \). Here \( x = 2 \), so \( b=2\sqrt{2} \).

Step2: Find the value of \( a \)

In a \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \).
The side opposite \( 30^{\circ} \) is the shortest side. If the side opposite \( 60^{\circ} \) is \( 3 \), and the hypotenuse \( a \) (opposite \( 90^{\circ} \)):
Using the ratio \( \frac{\text{hypotenuse}}{\text{side opposite }60^{\circ}}=\frac{2}{\sqrt{3}} \), so \( a=\frac{2\times3}{\sqrt{3}} = 2\sqrt{3} \).

Answer:

\( b = 2\sqrt{2} \)
\( a = 2\sqrt{3} \)