20.5 3D Pythagoras and trigonometry — Practise with help
Try each problem first. Open a hint if you need it, then use the working to check your method.
Warm-up: remember
Answer these from memory. Do not look back at the section.
- Find the hypotenuse of a right-angled triangle with shorter sides 9 and 12.
- Work out .
Answers
- . 2. .
If an answer differs, revisit Section 20.1 for Pythagoras' theorem and Section 19.1 for the area of a triangle.
Practise with help
1. A cuboid has base 3 by 4. Find its base diagonal.
Hint
Use Pythagoras on the base face.
Full working
b² = 3² + 4² = 25, so b = 5.
2. The base diagonal is 5 and height is 12. Find space diagonal.
Hint
Apply Pythagoras in the second triangle.
Full working
d² = 5² + 12² = 169, so d = 13.
3. A cuboid is 6 by 8 by 10. Find the angle between space diagonal and base.
Hint
First find the base diagonal, then use tangent.
Full working
Base diagonal = 10; tan θ = 10/10, so θ = 45°.
Before opening each full working, name the method and explain which measurements led you to choose it.