Right triangles and non-right triangles · Extension
| 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
- In a right triangle a²+b²=c²; sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent. For other triangles, use the sine or cosine rule, or area=ab sin C/2.
- 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°. With two sides 6 and 8 enclosing 60°, c²=6²+8²-2×6×8 cos60°=52.
Right triangles and non-right triangles
In a right triangle a²+b²=c²; sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent
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
- Use a plan or elevation for a three-dimensional problem before applying a triangle rule. Explain why the chosen triangle contains the required length or angle.
- 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 · Extension · 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.