Right triangles and non-right triangles
도입됨| English | 의미 |
|---|---|
| hypotenuse/haɪˈpɒtənjuːs/ | 합 |
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
- 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.
Which condition must hold before Pythagoras a²+b²=c² can be used?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Pythagoras is a property of right-angled triangles only, with c the hypotenuse. For other triangles the sine or cosine rule is needed.
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°. 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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Sine=opposite/hypotenuse=3/5=0.6.
Pythagoras applies to every triangle, including triangles without a right angle.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Pythagoras needs a right angle; for other triangles use the cosine rule.
Applying it
- 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.
Two sides 6 and 8 enclose 90°. Find their area.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
The sides are perpendicular, so area=6×8/2=24.
Match each ratio to its sides in a right-angled triangle.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
The sides are labelled relative to the chosen angle: opposite faces it, adjacent touches it, and the hypotenuse faces the right angle.
In your course
- Pearson Edexcel International A Level Mathematics, unit P1.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Hypotenuse: The side opposite the right angle in a right-angled triangle.
- In a right triangle a²+b²=c²; sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent.
- Watch for: label sides relative to the chosen angle.
Write from a hypotenuse of 17
- The legs are 8 and 15.
- Find the hypotenuse.
A right-angled triangle has legs 8 and 15. What is the hypotenuse?
289 has square root 17.
What is the hypotenuse?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
8-15-17 is a right-angled triple. Other options — 5 (the lesson hypotenuse); 23 (the legs added).