A right triangle has legs 5 and 12. Find the hypotenuse.
Answer: 13
Reasoning: c=√(5²+12²)=13.
Reasoning: c=√(5²+12²)=13.
Unit 06 · Lesson 08 · Right Triangles, Trigonometry & Bearings
Bearings, Vectors & Multi-Step Navigation · Teacher Key
Translate compass bearings into right triangles or vectors and combine multiple legs of travel.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to translate compass bearings into right triangles or vectors and combine multiple legs of travel.
Success evidence
- Label the figure and units before calculating.
- Select and justify the governing definition, theorem, relationship, or construction.
- Show enough work that another learner can audit the conclusion.
- Check the result against the diagram, scale, and stated constraints.
Visual model
Use right-triangle relationships and trigonometric ratios to determine lengths, angles, heights, bearings, and directions.
a^2+b^2=c^2\sin\theta=\frac{opposite}{hypotenuse}\cos\theta=\frac{adjacent}{hypotenuse}\tan\theta=\frac{opposite}{adjacent}
Worked reasoning
Process: Sketch and label the triangle, identify the known and unknown sides relative to the reference angle, choose one ratio, solve, and interpret units.
- Read the figure and rewrite all given information with correct notation.
- Identify the exact relationship that connects the known and unknown quantities.
- Substitute, construct, or prove one justified step at a time.
- State the result with units and a sentence explaining what it means.
Misconception check: Opposite and adjacent depend on the chosen reference angle; the hypotenuse is always opposite the right angle.
Khaemenes Academy · Open-Age Geometry
Bearings, Vectors & Multi-Step Navigation
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
In a 45-45-90 triangle with leg 6, find the hypotenuse.
Answer: 6√2
Reasoning: The side ratio is 1:1:√2.
Reasoning: The side ratio is 1:1:√2.
In a 30-60-90 triangle with short leg 5, find the long leg.
Answer: 5√3
Reasoning: The side ratio is 1:√3:2.
Reasoning: The side ratio is 1:√3:2.
From a 35° angle, the hypotenuse is 17. Find the opposite side to the nearest tenth.
Answer: 9.8
Reasoning: opposite=17·sin(35°)≈9.8.
Reasoning: opposite=17·sin(35°)≈9.8.
If sin θ=0.8, find θ to the nearest degree.
Answer: 53°
Reasoning: Use inverse sine.
Reasoning: Use inverse sine.
A bearing of N 35° E is measured from which reference direction?
Answer: North, turning 35° toward east
Reasoning: Quadrant bearings begin at north or south.
Reasoning: Quadrant bearings begin at north or south.