Question

for the right triangles below, find the values of the side lengths d and h. round your answers to the nearest tenth. (a) d = (b) h =

Explanation:

Step1: Find \( d \) in the 30 - 60 - 90 triangle

In a 30 - 60 - 90 right triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \), where the side opposite \( 30^{\circ} \) is the shortest one, the side opposite \( 60^{\circ} \) is \( \sqrt{3} \) times the shortest side, and the hypotenuse is twice the shortest side. Here, the side adjacent to \( 60^{\circ} \) (the side with length 7) is opposite \( 30^{\circ} \)? Wait, no. Wait, the right angle is between \( d \) and the side of length 7. So the angle of \( 30^{\circ} \) is at the top, so the side opposite \( 30^{\circ} \) is \( d \), and the side opposite \( 60^{\circ} \) is 7. Wait, no, let's use trigonometry. We know that \( \tan(60^{\circ})=\frac{\text{opposite}}{\text{adjacent}}=\frac{7}{d} \). Since \( \tan(60^{\circ}) = \sqrt{3}\approx1.732 \), we can solve for \( d \):

\( d=\frac{7}{\tan(60^{\circ})}=\frac{7}{\sqrt{3}}\approx\frac{7}{1.732}\approx4.0 \) (Wait, no, wait. Wait, the angle at the bottom is \( 60^{\circ} \), so the side opposite \( 60^{\circ} \) is the vertical side (length 7), and the side adjacent to \( 60^{\circ} \) is \( d \). So \( \tan(60^{\circ})=\frac{\text{opposite}}{\text{adjacent}}=\frac{7}{d} \), so \( d = \frac{7}{\tan(60^{\circ})}=\frac{7}{\sqrt{3}}\approx4.0 \)? Wait, no, \( \tan(60^{\circ})=\sqrt{3}\approx1.732 \), so \( d=\frac{7}{1.732}\approx4.0 \)? Wait, or maybe I mixed up. Wait, the angle of \( 30^{\circ} \) is at the top, so the side opposite \( 30^{\circ} \) is \( d \), and the side opposite \( 60^{\circ} \) is 7. So in a 30 - 60 - 90 triangle, the side opposite \( 30^{\circ} \) is \( x \), the side opposite \( 60^{\circ} \) is \( x\sqrt{3} \), and hypotenuse is \( 2x \). So if the side opposite \( 60^{\circ} \) is 7, then \( x\sqrt{3}=7 \), so \( x = \frac{7}{\sqrt{3}}\approx4.0 \), which is \( d \). So \( d\approx4.0 \).

Step2: Find \( h \) in the 45 - 45 - 90 triangle

In a 45 - 45 - 90 right triangle, the legs are equal, and the hypotenuse \( h \) is \( \text{leg}\times\sqrt{2} \). Here, one leg is 5, so the hypotenuse \( h = 5\sqrt{2}\approx5\times1.414\approx7.1 \).

Answer:

(a) \( d\approx4.0 \)
(b) \( h\approx7.1 \)