Question

for the right triangles below, find the exact values of the side lengths b and c. if necessary, write your responses in simplified radical form. 30° 8 b 60° c 45° 7 45° b = c =

Explanation:

Step1: Find \( b \) (30-60-90 triangle)

In a 30-60-90 right triangle, the sides are in the ratio \( 1 : \sqrt{3} : 2 \) (shorter leg : longer leg : hypotenuse). The longer leg (opposite 60°) is \( 8 \), and the shorter leg \( b \) (opposite 30°) relates to the longer leg by \( \text{longer leg} = b\sqrt{3} \). So, \( 8 = b\sqrt{3} \)? Wait, no—wait, angle 60° is at the base, so the side adjacent to 60° is \( b \), opposite is 8? Wait, no, the right angle is at the bottom right, so the vertical side is 8 (adjacent to 30°), horizontal is \( b \) (opposite to 30°). Wait, 30° angle: in 30-60-90, the side opposite 30° is the shortest leg, opposite 60° is longer leg, hypotenuse is twice the shortest leg. So angle 30°: opposite side is \( b \), adjacent is 8, hypotenuse is \( 2b \). Then, \( \tan(30^\circ) = \frac{b}{8} \), so \( b = 8 \tan(30^\circ) \). \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so \( b = \frac{8}{\sqrt{3}} = \frac{8\sqrt{3}}{3} \)? Wait, no, wait the triangle: angle 60° at the bottom left, right angle at bottom right, so the vertical side is 8 (adjacent to 60°), horizontal side \( b \) (opposite to 60°). So \( \tan(60^\circ) = \frac{8}{b} \), so \( b = \frac{8}{\tan(60^\circ)} = \frac{8}{\sqrt{3}} = \frac{8\sqrt{3}}{3} \)? Wait, no, maybe I mixed up. Wait, 30-60-90 triangle: angles 30, 60, 90. The side opposite 30° is the shortest, let's call it \( x \), opposite 60° is \( x\sqrt{3} \), hypotenuse \( 2x \). In the first triangle, the angle at the top is 30°, so the side opposite 30° is \( b \) (horizontal), and the side opposite 60° is 8 (vertical). So \( 8 = x\sqrt{3} \), \( b = x \). So \( x = \frac{8}{\sqrt{3}} = \frac{8\sqrt{3}}{3} \)? Wait, no, that can't be. Wait, maybe the vertical side is the longer leg (opposite 60°), so \( \text{longer leg} = \text{shorter leg} \times \sqrt{3} \). So shorter leg (opposite 30°) is \( b \), longer leg (opposite 60°) is 8. So \( 8 = b\sqrt{3} \implies b = \frac{8}{\sqrt{3}} = \frac{8\sqrt{3}}{3} \)? Wait, maybe I made a mistake. Alternatively, using trigonometry: \( \tan(30^\circ) = \frac{b}{8} \implies b = 8 \tan(30^\circ) = 8 \times \frac{1}{\sqrt{3}} = \frac{8\sqrt{3}}{3} \).

Step2: Find \( c \) (45-45-90 triangle)

In a 45-45-90 right triangle, the legs are equal, and the hypotenuse \( c \) is \( \text{leg} \times \sqrt{2} \). The leg is 7, so \( c = 7\sqrt{2} \).

Wait, wait, let's recheck the first triangle. The first triangle has angles 30°, 60°, 90°. The right angle is at the bottom right, so the sides: horizontal is \( b \) (adjacent to 30°), vertical is 8 (opposite to 30°). Wait, no: angle at the top is 30°, so the angle between hypotenuse and vertical side is 30°, so vertical side is adjacent to 30°, horizontal is opposite. So \( \tan(30^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{8} \implies b = 8 \tan(30^\circ) = 8 \times \frac{1}{\sqrt{3}} = \frac{8\sqrt{3}}{3} \). Correct.

For the second triangle: 45-45-90, legs are equal (both 7), hypotenuse \( c = 7\sqrt{2} \).

Answer:

\( b = \frac{8\sqrt{3}}{3} \), \( c = 7\sqrt{2} \)