Right triangles and non-right triangles · Higher · 直角三角形与非直角三角形 · 高阶
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| 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? · 哪个描述正确定义了斜边？
The side opposite the right angle in a right-angled triangle. · 直角三角形中直角所对的边。
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 · 直角三角形与非直角三角形
In a right triangle a²+b²=c²; sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent · 在直角三角形中，a²+b²=c²；sinθ=对边/斜边，cosθ=邻边/斜边，tanθ=对边/邻边。
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find the hypotenuse when the legs are 3 and 4. · 已知两直角边分别为3和4，求斜边长度。
Pythagoras gives √(3²+4²)=5. · 根据勾股定理，√(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. · 已知对边=3，斜边=5，求sinθ的值。
Sine=opposite/hypotenuse=3/5=0.6. · 正弦=对边/斜边=3/5=0.6。
Pythagoras applies to every triangle, including triangles without a right angle. · 勾股定理适用于所有三角形，包括非直角三角形。
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. · 相对于所选角标注各边。勾股定理需要直角存在。计算器角度模式错误可能导致看似合理但实际错误的结果。后续步骤中请保留未舍入的数值。
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. · 已知两边长分别为6和8，夹角为90°。求其面积。
The sides are perpendicular, so area=6×8/2=24. · 因两边互相垂直，故面积=6×8/2=24。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- 4MA1 · Higher · 4. 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.