Right triangles and non-right triangles
Введено| English | Значение |
|---|---|
| hypotenuse/haɪˈpɒtənjuːs/ | гипотенуза |
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?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
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.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
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.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Sine=opposite/hypotenuse=3/5=0.6.
Pythagoras applies to every triangle, including triangles without a right angle.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
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.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The sides are perpendicular, so area=6×8/2=24.
Match each ratio to its sides in a right-angled triangle.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
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
- AQA A-level Mathematics (7357), specification section E.
- 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.