Right triangles and non-right triangles · Core
| English | 中文 | Pinyin |
|---|---|---|
| hypotenuse/haɪˈpɒtənjuːs/ | 斜边 | xié biān |
Which side does the ladder need?
- A ladder reaches a height of 4 m while its foot is 3 m from a wall. Which sides are known, and which angle do we need?
- This lesson studies hypotenuse 斜边: The side opposite the right angle in a right-angled triangle.
Choose the mathematical structure
- Use Pythagoras in a right triangle and use sine, cosine or tangent with the sides labelled relative to the chosen angle.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines hypotenuse?
The side opposite the right angle in a right-angled triangle.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
The ladder length is c=√(3²+4²)=5 m. Its angle to the ground satisfies tanθ=4/3, so θ≈53.1°. A right triangle with legs 6 and 8 has area 6×8/2=24.
Right triangles and non-right triangles
Use Pythagoras in a right triangle and use sine, cosine or tangent with the sides labelled relative to the chosen angle
Compare the model with the worked case and explain one change.
Find the hypotenuse when the legs are 3 and 4.
Pythagoras gives √(3²+4²)=5.
Test a tempting shortcut
- Label sides relative to the chosen angle. Pythagoras needs a right angle. A calculator angle mode error can produce a plausible but wrong result. Keep unrounded values for later steps.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Pythagoras applies to every triangle, including triangles without a right angle. This claim is false. Explain which definition or assumption it violates.
Find sinθ when opposite=3 and hypotenuse=5.
Sine=opposite/hypotenuse=3/5=0.6.
Pythagoras applies to every triangle, including triangles without a right angle.
Label sides relative to the chosen angle. Pythagoras needs a right angle. A calculator angle mode error can produce a plausible but wrong result. Keep unrounded values for later steps.
Interpret a new situation
- This Foundation/Core lesson uses right-angled triangles only. Sine and cosine rules for non-right triangles belong to the advanced-tier lesson.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Two sides 6 and 8 enclose 90°. Find their area.
The sides are perpendicular, so area=6×8/2=24.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 9260 · Core · 3.3. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The side opposite the right angle in a right-angled triangle. Choose the relationship, show the method, check its assumptions and interpret the result.