Applied Right Triangle Scenarios
Section 7.8 -- Applied right triangle problems
Step 1 of 5: Read the scene, draw the picture
"A person standing 200 feet from a cell tower measures a 62-degree angle of elevation to the top." No picture yet. Where does each piece go?
Click each zone to place the element in the right spot.
Place these elements (click the correct zone for each):
Step 2 of 5: Where does the angle go?
An angle of elevation is measured UP from the horizontal. Where do you place the 62-degree angle?
Step 3 of 5: Angle of depression
Now you are at the top of the tower, looking down at the person. Where is the angle of depression measured from?
Step 4 of 5: Label the triangle
You have a 62-degree angle at the observer. You know the distance along the ground (200 ft). You want the tower height. Relative to your 62-degree angle, what is the 200 ft side and what is the tower height?
The 200 ft side (along the ground) is:
Step 5 of 5: Pick the ratio and solve
You know the adjacent side (200 ft) and want the opposite side (h). Which trig ratio connects these two sides?
Scenario Sandbox
Set up your own right triangle scenario. Adjust two values and the third resolves automatically.
Move the sliders to see how angle and distance determine height. Toggle the depression view to see the alternate-interior-angle relationship.
Try This
Height of a Cell Tower
A person stands 200 feet from the base of a cell tower and measures a 62-degree angle of elevation to the top. How tall is the tower?
The setup: \(\tan(62°) = \dfrac{h}{200}\), so \(h = 200 \cdot \tan(62°)\).
Answer within 1 foot. Correct answer: approximately 376.2 ft.
Lifeguard's View
A lifeguard sits in a chair with her eyes 8 feet above the water. She spots a swimmer at a 15-degree angle of depression. How far out from the base of her chair is the swimmer?
The angle of depression from the horizontal equals the angle of elevation from the swimmer. So: \(\tan(15°) = \dfrac{8}{d}\), meaning \(d = \dfrac{8}{\tan(15°)}\).
Answer within 1 foot. Correct answer: approximately 29.9 ft.
Tree Shadow
A tree casts a 45-foot shadow when the sun is 38 degrees above the horizon. How tall is the tree?
\(\tan(38°) = \dfrac{h}{45}\), so \(h = 45 \cdot \tan(38°)\).
Answer within 1 foot. Correct answer: approximately 35.2 ft.
Boat Navigation
A boat sails on bearing 035 for 12 nautical miles, then turns to bearing 125 for 8 nautical miles. Bearing 035 and bearing 125 differ by exactly 90 degrees, so the two legs form a right angle. How far is the boat from its starting point?
The two legs are perpendicular, so use the Pythagorean theorem: \(d = \sqrt{12^2 + 8^2}\).
Answer within 0.5 nm. Correct answer: approximately 14.4 nm.
Roof Pitch
A roof has a 6/12 pitch: it rises 6 inches for every 12 inches of horizontal run. What angle does that make with the horizontal? The house is 32 feet wide and the ridge runs down the center. How long is each rafter?
The pitch 6/12 means a rise of 6 inches per 12 inches of run, which simplifies to a ratio of 1/2. So \(\tan(\theta) = \frac{6}{12} = 0.5\), giving \(\theta = \arctan(0.5)\).
The house is 32 ft wide, so each side runs 16 ft horizontally. With 6/12 pitch, the rise is \(16 \times \frac{6}{12} = 8\) ft. The rafter length is the hypotenuse: \(\sqrt{16^2 + 8^2}\).
Answer within 1 degree. Correct answer: approximately 26.6°.
Answer within 0.5 feet. Correct answer: approximately 17.9 ft.
- Translate a word problem into a labeled right triangle diagram
- Distinguish angles of elevation from angles of depression and identify them correctly
- Label triangle sides as opposite, adjacent, or hypotenuse relative to a given angle
- Select the correct trig ratio based on which sides are known and unknown
- Solve applied right triangle problems including height/distance, navigation, and construction scenarios
Quick Check
A surveyor stands 150 feet from the base of a building and measures a 70-degree angle of elevation to the roof. Which equation correctly finds the building height \(h\)?
Instructor Notes
Teaching Notes
This simulation opens in Apply mode by default because the underlying concept (right triangle trig ratios) was built in Simulation 7. The Discover sequence here focuses on the setup strategy -- translating a word problem into a picture -- rather than deriving trig ratios from scratch.
The five Discover steps walk through a single problem end to end. The key insight is Step 5: you do not choose a trig ratio by guessing. You choose it by labeling the triangle relative to your angle and reading off which ratio connects your known to your unknown.
Common Student Errors
- Measuring the angle of elevation from the vertical instead of from the horizontal. Step 2 directly addresses this with a visual comparison.
- Confusing angle of depression with angle of elevation. Step 3 shows they are alternate interior angles and equal.
- Forgetting eye height or instrument height. The starter problem notes this as a follow-up consideration.
- Choosing the wrong trig ratio because they do not label sides relative to the angle first.
Discussion Questions
- Why does the angle of depression from the top equal the angle of elevation from the bottom? Can you prove this using parallel lines?
- In the cell tower problem, the person's eyes are about 5.5 feet above the ground. How does this affect the answer? By how much?
- A surveyor measures two angles of elevation from two different distances. How could they find a height without knowing any side length directly?
Exam Connection
Applied right triangle problems are common on exams. Students typically see one or two word problems requiring angle of elevation/depression setup. The most frequent mistakes on exams are drawing the angle in the wrong location and picking the wrong trig ratio. Both are addressed directly in the Discover sequence.