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Solving Right Triangles | Part 2 of 2

Solving Right Triangles

Part 2 of 2: Applications

In this lesson:

  • Solve problems with angles of elevation and depression
  • Apply right triangles to navigation and bearings
  • Use the "draw first, solve second" approach
High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Learning Objectives

By the end of this lesson, you should be able to:

  1. Solve real-world problems using angles of elevation and depression
  2. Apply right triangle solving to navigation and bearings
  3. Draw accurate diagrams before solving
High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

How Tall Is That Building?

You can measure it - without climbing.

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Angle of Elevation

Progressive diagram showing person looking up at building with horizontal reference line at eye level and angle marked upward

Measured upward from the horizontal at eye level

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Angle of Depression

Progressive diagram showing person on cliff looking down at boat with horizontal reference line at eye level and angle marked downward

Measured downward from the horizontal at eye level

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Key Relationship

Diagram showing observer at top and bottom of a cliff with parallel horizontal lines, marking angle of depression from top equals angle of elevation from bottom with alternate interior angles highlighted

Elevation from bottom = Depression from top (alternate interior angles)

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Check-In

Where is the horizontal reference - on the ground or at eye level?

Think about why this matters for drawing your diagram...

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 1: Building Height

Diagram showing person 50m from building base measuring 60° angle of elevation to the top with right triangle clearly marked and sides labeled

Given: 50 m from building, 60° elevation

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 1: Solution

Find: height of building

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 2: Distance to Boat

Diagram showing person on 100m cliff with 25° angle of depression to boat on water with right triangle marked

Given: 100 m cliff, 25° depression to boat

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 2: Solution

Find: horizontal distance to boat

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Check-In

In Example 2, where exactly is the right angle in your diagram?

Locate it before moving on...

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 3: Line-of-Sight Distance

Diagram showing airplane at 5000ft altitude with observer on ground measuring 40° elevation with line-of-sight distance marked as unknown

Given: airplane at 5000 ft, 40° elevation

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 3: Solution

Find: line-of-sight distance

Why sine, not tangent? Because we're connecting opposite to hypotenuse.

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Key Tips

When solving elevation and depression problems:

  1. Draw first, solve second - always
  2. Horizontal reference is at eye level, not the ground
  3. Label O, A, H relative to the angle you're using
  4. Check: Does your answer make sense? (e.g., building height ≈ reasonable)
High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Transition

Same tools, new context - from looking up and down to navigating across

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Bearing Notation

Compass diagram with cardinal directions and several bearings plotted: N30°E shown as starting at north and rotating 30° toward east

N30°E = "start at north, rotate 30° toward east"

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Compass Practice

Compass diagram showing multiple bearings plotted: N30°E, S45°W, N60°W, and bearing 090° (due east)

Practice reading: N30°E, S45°W, N60°W, bearing 090°

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Check-In

Draw N40°E on a compass. Which direction do you start from?

Visualize it before moving on...

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 1: Hiker Distance and Bearing

Grid showing hiker's path: 4 km east then 3 km north with right triangle formed and direct distance marked

Given: 4 km east, then 3 km north

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 1: Solution

Find: distance and bearing

Distance: km

Bearing angle: → N53.1°E

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 2: Component Breakdown

Diagram showing plane flying at bearing N30°E for 200 km with north and east components marked

Given: plane flies N30°E for 200 km

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 2: Solution

Find: north and east components

North: km

East: km

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Check-In

For bearing N30°E - which component uses cosine: north or east?

Think about opposite, adjacent, and hypotenuse...

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 3: Multi-Leg Journey

Grid showing ship sailing 10 km north, 15 km east, 5 km south with net displacement and bearing marked

Given: 10 km N, 15 km E, 5 km S

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Example 3: Solution

Find: displacement distance and bearing

Net: N = 10 − 5 = 5 km, E = 15 km

Distance: km

Bearing: → N71.6°E

High School Geometry | HSG.SRT.C.8
Solving Right Triangles | Part 2 of 2

Summary

"Draw first, solve second" - right triangles are everywhere

✓ Elevation and depression problems: horizontal reference at eye level
✓ Navigation problems: break into N/S and E/W components
✓ Same tools: trig ratios, inverse trig, Pythagorean Theorem

You can now solve right triangles in the real world.

High School Geometry | HSG.SRT.C.8