A right triangle has legs 3 and 4. Find the hypotenuse.
Answer: 5
Reasoning: c=√(3²+4²)=5.
Reasoning: c=√(3²+4²)=5.
Unit 06 · Lesson 04 · Right Triangles, Trigonometry & Bearings
Geometric Mean & Right-Triangle Altitudes · Teacher Key
Use altitude-to-hypotenuse similarity and geometric mean relationships.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to use altitude-to-hypotenuse similarity and geometric mean relationships.
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
Geometric Mean & Right-Triangle Altitudes
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 16. Find the opposite side to the nearest tenth.
Answer: 9.2
Reasoning: opposite=16·sin(35°)≈9.2.
Reasoning: opposite=16·sin(35°)≈9.2.
If sin θ=0.75, find θ to the nearest degree.
Answer: 49°
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.