Right-angled triangles in two dimensions · Foundation
導入済み| English | 意味 |
|---|---|
| hypotenuse/haɪˈpɒtənjuːs/ | 斜辺 |
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.
The method
- Use Pythagoras in a right triangle and use sine, cosine or tangent with the sides labelled relative to the chosen angle.
A ladder forms a right triangle with legs 3 m and 4 m. How long is the ladder?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
By Pythagoras, ladder²=3²+4²=25, so the ladder is 5 m. Adding the legs ignores the right-angle structure.
A worked case
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-angled triangles in two dimensions
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Pythagoras gives √(3²+4²)=5.
Where it goes wrong
- 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.
Not true: Pythagoras applies to every triangle, including triangles without a right angle. Pythagoras needs a right angle; for other triangles use the cosine rule.
Find sinθ when opposite=3 and hypotenuse=5.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Sine=opposite/hypotenuse=3/5=0.6.
Pythagoras applies to every triangle, including triangles without a right angle.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Pythagoras needs a right angle; for other triangles use the cosine rule.
Applying it
- This Foundation/Core lesson uses right-angled triangles only.
- Sine and cosine rules for non-right triangles belong to the advanced-tier lesson.
Two sides 6 and 8 enclose 90°. Find their area.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
The sides are perpendicular, so area=6×8/2=24.
Match each side name to its position in a right-angled triangle.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Sides are named relative to the angle being used; the hypotenuse is always opposite the right angle.
In your course
- AQA GCSE Mathematics (8300), Foundation tier, specification section 3.4.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Hypotenuse: The side opposite the right angle in a right-angled triangle.
- Use Pythagoras in a right triangle and use sine, cosine or tangent with the sides labelled relative to the chosen angle.
- Watch for: label sides relative to the chosen angle.