Question

determining an angle of depression
use the diagram and complete the steps to find the measure of the angle of depression from the top of the hoop to lisa.
the length of the shortest leg of the right triangle that is formed is ▼ feet.
the angle of depression from the hoop to lisa is ▼ to the angle of elevation from lisas line of sight to the hoop.
because the lengths of the opposite and adjacent sides are known, use the ✔ inverse tangent ▼ function.
the angle of depression, rounded to the nearest degree, is approximately ▼ degrees.

Explanation:

Step1: Identify the shortest leg

The two legs of the right - triangle are \(5\) ft and \(8.5\) ft. Since \(5<8.5\), the length of the shortest leg is \(5\) feet.

Step2: Relationship between angles of depression and elevation

The angle of depression from the hoop to Lisa is congruent (equal) to the angle of elevation from Lisa’s line of sight to the hoop. This is because of the property of parallel lines (the horizontal line from the hoop and the horizontal line at Lisa’s eye - level are parallel, and the line of sight is a transversal).

Step3: Use the inverse tangent function

We know that for a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Let \(\theta\) be the angle of depression (or elevation). Here, the opposite side to the angle \(\theta\) is \(8.5\) ft and the adjacent side is \(5\) ft. So, \(\theta = \tan^{- 1}(\frac{8.5}{5})\).
Calculate \(\frac{8.5}{5}=1.7\). Then \(\theta=\tan^{-1}(1.7)\). Using a calculator, \(\tan^{-1}(1.7)\approx59.53^{\circ}\approx60^{\circ}\)

Answer:

The length of the shortest leg of the right triangle that is formed is \(5\) feet. The angle of depression from the hoop to Lisa is congruent (equal) to the angle of elevation from Lisa’s line of sight to the hoop. The angle of depression, rounded to the nearest degree, is approximately \(60\) degrees.