📚 Solving a Ladder Problem with Pythagoras' Theorem | 用毕达哥拉斯定理解梯子问题
In this article, we work through Question 1 from page 225 of the Cambridge KS3 Mathematics coursebook, which applies Pythagoras' theorem to a practical ladder problem. Understanding how to set up a right-angled triangle from a real-life situation is a key skill at this level.
在本文中，我们将讲解剑桥 KS3 数学教材第 225 页的第 1 题，该题将毕达哥拉斯定理应用于一个实际的梯子问题。学会从实际情境中构建直角三角形是本阶段的重要技能。
1. The Ladder Problem | 梯子问题
The problem states: A ladder 5 m long leans against a vertical wall. The foot of the ladder is placed 2 m away from the base of the wall. Calculate how high up the wall the ladder reaches. This situation creates a right-angled triangle, with the ladder forming the hypotenuse.
题目描述：一把长 5 米的梯子靠在竖直的墙上，梯脚距墙脚 2 米。求梯子顶端距离地面的高度。这一情境形成了一个直角三角形，梯子构成斜边。
2. Identifying the Sides of the Triangle | 识别三角形的边
In any right-angled triangle, the hypotenuse is the side opposite the right angle and is always the longest side. Here, the ladder length (5 m) is the hypotenuse (c). The horizontal distance from the wall to the foot of the ladder, 2 m, is one leg (b). The height up the wall is the unknown leg (a).
在任何直角三角形中，斜边是直角对着的边，并且总是最长的边。在此例中，梯子长度（5 米）为斜边（c）。梯脚距墙的水平距离 2 米是一条直角边（b）。梯子顶端沿墙的高度是未知的另一条直角边（a）。
3. Applying Pythagoras' Theorem | 应用毕达哥拉斯定理
Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Written as an equation:
毕达哥拉斯定理指出，在直角三角形中，斜边的平方等于两条直角边的平方和。公式如下：
a² + b² = c²
Substituting the known values (c = 5 m, b = 2 m) gives:
代入已知值（c = 5 米，b = 2 米）得：
a² + 2² = 5²
4. Solving for the Unknown Height | 求解未知高度
4 and 5² = 25. The equation becomes a² + 4 = 25. Subtract 4 from both sides to isolate a²:
首先计算平方：2² = 4，5² = 25。方程变为 a² + 4 = 25。两边同时减去 4，得到 a²：
a² = 25 − 4 = 21
To find the value of a, take the square root of both sides. The exact answer is a = √21 metres.
要得到 a 的值，需要对两边开平方。精确答案为 a = √21 米。
5. Finding an Approximate Value | 求近似值
Surds like √21 often need to be converted to a decimal for practical use. Using a calculator, √21 mon to round to two decimal places: a ≈ 4.58 m. Always state the degree of rounding, e.g., “to 2 d.p.”.
像 √21 这样的根式在实际应用中常需转换为小数。用计算器可得 √21 ≈ 4.582575694…… 在 KS3 阶段，通常四舍五入到两位小数：a ≈ 4.58 m。务必标注近似程度，例如“保留两位小数”。
6. Providing the Final Answer with Units | 给出带单位的最终答案
Remember that measurements must include units. The exact answer can be left as √21 m, but the approximate answer is 4.58 m (to 2 d.p.). Both forms are acceptable, but you should check which form the question expects.
请记住，测量值必须注明单位。精确答案可写成 √21 m，近似答案则为 4.58 m（保留两位小数）。两种形式均可，但需注意题目要求的形式。
7. Verifying Your Solution | 验证你的解
. This confirms the calculation is correct. Also consider whether the answer is physically feasible – ight up the wall (approx. 4.58 m) is less than the ladder length (5 m), which makes sense.
快速验证有助于确认答案合理。将近似直角边长平方并相加：4.58² ≈ 20.98，加上 2² = 4，得到约 24.98，十分接近 25（因为 5² = 25）。这证明计算正确。同时考虑实际情况是否可行——梯子顶端高度（约 4.58 m）小于梯子长度（5 m），这符合直觉。
8. Common Errors to Avoid | 避免常见错误
One frequent mistake is misidentifying the hypotenuse. Always check that the hypotenuse is opposite the right angle and is the longest side. Do not assume the leaning object is automatically the hypotenuse without drawing a diagram. Another error is forgetting to square the numbers correctly: 5² is 25, not 10. Also, when rearranging, ensure you subtract, not add. Finally, always include units in your final answer.
一个常见错误是混淆斜边。务必确认斜边对着直角且为最长边。不应在没有画图的情况下就默认倾斜物体即为斜边。另一个错误是平方计算有误：5² 等于 25 而非 10。移项时也要注意是减法而非加法。最后，一定在最终答案中带上单位。
9. Understanding Exact and Approximate Forms | 理解精确值与近似值
Exact surd form (√21) is mathematically precise and often required in further calculations to avoid rounding errors. The approximate decimal (4.58) is more meaningful in real-world measurements. KS3 students should be comfortable leaving answers as surds and rounding to a specified number of decimal places.
精确的根式形式（√21）在数学上更严谨，且在后续计算中常用以避开舍入误差。近似的小数（4.58）在实际测量中更具象。KS3 学生应熟练掌握保留根式形式的答案，以及按要求四舍五入到指定小数位。
10. Extending the Skill: Other Right-Angled Contexts | 技能延伸：其他直角三角形情境
This ladder problem is just one example. Pythagoras' to draw a clear right-angled triangle and label the known sides before using a² + b² = c².
这个梯子问题只是一个例子。毕达哥拉斯定理还广泛应用于计算矩形对角线长度、坐标网格上两点间距离、圆锥的斜高，或建筑中支撑臂的边长等。每种情形的关键步骤都是先画出清晰的直角三角形并标注已知边长，然后再运用 a² + b² = c²。
11. Key Takeaways | 要点总结
Identify the hypotenuse (longest side, opposite the right angle) and the two legs.
识别斜边（最长边，正对直角）和两条直角边。
formula a² + b² = c² and substitute carefully, paying attention to which side is unknown.
写出公式 a² + b² = c² 并仔细代入，注意哪条边是未知边。
Solve for the unknown side using square roots, and decide whether to leave the answer as a surd or give a rounded decimal.
使用平方根求未知边长，并决定是保留根式形式还是给出四舍五入的小数。
Check your answer and always include units.
检查答案并始终带上单位。
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