Right triangles and non-right triangles · Extension
הוצג| English | משמעות |
|---|---|
| hypotenuse/haɪˈpɒtənjuːs/ | יתר |
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?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Sine=opposite/hypotenuse=3/5=0.6.
Pythagoras applies to every triangle, including triangles without a right angle.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
The sides are perpendicular, so area=6×8/2=24.
Match each ratio to its sides in a right-angled triangle.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
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
- OxfordAQA International GCSE Mathematics (9260), Extension tier, specification section 3.3.
- 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 third side of 14
- Two sides of 6 and 10 enclose an angle of 120 degrees.
- Find the third side. Cosine of 120 degrees is negative.
Two sides of lengths 6 and 10 enclose an angle of 120 degrees. What is the third side?
36 plus 100 plus 60 is 196, and 14 squared is 196.
What is the third side?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The cosine rule gives 196, whose square root is 14.