📚 Pythagoras' Theorem: Finding Missing Sides in Right-Angled Triangles | 勾股定理：求直角三角形中的未知边长
In Cambridge KS3 mathematics, Pythagoras' theorem is one of the most powerful tools for solving problems involving right-angled triangles. It links the three sides of a right-angled triangle through a simple square relationship. This article explains the theorem, shows how to find a missing hypotenuse or a shorter side, and gives you exam-ready practice tips.
在剑桥 KS3 数学中，勾股定理是解决直角三角形问题最有力的工具之一。它通过简单的平方关系将直角三角形的三条边联系起来。本文将解释这一定理，展示如何求未知的斜边或直角边，并给出备考练习建议。
1. What is Pythagoras' theorem? | 什么是勾股定理？
Pythagoras' theorem applies only to right-angled triangles. If we label the longest side, opposite the right angle, as the hypotenuse c, and the other two shorter sides as a and b, then the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
勾股定理只适用于直角三角形。如果我们把最长的边（直角的对边）标为斜边 c，另外两条较短的边标为 a 和 b，那么斜边上的正方形面积等于另外两条边上的正方形面积之和。
a² + b² = c²
This equation is the standard form used in Cambridge Checkpoint questions. Always check that the angle opposite the side you call c is 90° before applying the formula.
这个等式是剑桥 Checkpoint 题目中使用的标准形式。在应用公式之前，务必确认你所标的 c 边所对的角是 90°。
2. Labelling a right-angled triangle | 标记直角三角形
Start by identifying the right angle. The side opposite the right angle is always the hypotenuse and is the longest side. The two sides that form the right angle are called the legs, or shorter sides.
首先找出直角。直角的对边总是斜边，也是最长的边。构成直角的两条边称为直角边或较短边。
In some questions, the triangle is drawn in different orientations, so the hypotenuse is not always at the top or sloping in the same direction. Do not assume a side is the hypotenuse just because it looks longest; check it is opposite the marked right angle.
在某些题目中，三角形的绘制方向不同，斜边不一定总在上方或按同一方向倾斜。不要仅仅因为某条边看起来最长就认为它是斜边；要检查它是否对着标记出的直角。
3. Finding the hypotenuse | 求斜边
sides are known, substitute into a² + b² = c², add the squares, then take the square root.
当两条直角边已知时，将它们代入 a² + b² = c²，先求平方和，再开平方。
Example: Find c if a = 6 cm and b = 8 cm.
例：已知 a = 6 厘米，b = 8 厘米，求 c。
c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm
Notice that the answer must be positive because it is a length. If you get a negative square root, ignore it in geometry contexts.
注意答案必须为正数，因为它是长度。如果你得到负的平方根，在几何情境中应舍去。
4. Finding a shorter side | 求较短的直角边
the formula. If c and b are known, then a² = c² − b². Take the square root of the result.
要求未知的较短直角边，需要变形公式。如果已知 c 和 b，那么 a² = c² − b²。对结果开平方即可。
Example: Find a if c = 13 cm and b = 5 cm.
例：已知 c = 13 厘米，b = 5 厘米，求 a。
a² = 13² − 5² = 169 − 25 = 144, so a = √144 = 12 cm
ctise writing the subtraction form directly rather than always starting with a² + b² = c².
这种变形经常出现在多步计算题中，因此要练习直接写出减法形式，而不是每次总是从 a² + b² = c² 开始。
5. Pythagorean triples | 勾股数
-5, 5-12-13, 7-24-25 and 8-15-17. You may also see multiples of these, such as 6-8-10 or 9-12-15.
勾股数是满足 a² + b² = c² 的三个正整数。最常见的勾股数有 3-4-5、5-12-13、7-24-25 和 8-15-17。你也可能见到它们的倍数，例如 6-8-10 或 9-12-15。
Recognising triples can save time in non-calculator Cambridge Checkpoint papers. If two sides of a right-angled triangle are 6 and 8, the hypotenuse is 10 without any calculation.
识别勾股数可以在剑桥 Checkpoint 非计算器试卷中节省时间。如果直角三角形的两条边分别是 6 和 8，那么不用计算即可知道斜边是 10。
6. The converse of Pythagoras' theorem | 勾股定理的逆定理
+ b² = c² holds, then the triangle is right-angled, with the right angle opposite side c.
逆定理指出，如果一个三角形的三边为 a、b 和 c，并且满足 a² + b² = c²，那么这个三角形是直角三角形，直角就在边 c 的对面。
Exam questions may ask you to show whether a triangle is right-angled. Substitute the three side lengths into the equation, with the longest side treated as c. If both sides of the equation are equal, the triangle is right-angled; if not, it is not.
考试题可能要求你判断一个三角形是否为直角三角形。将三条边长代入等式，将最长边看作 c。如果等式两边相等，则三角形是直角三角形；如果不相等，则不是。
7. Applying Pythagoras' theorem to word problems | 勾股定理在应用题中的使用
Many KS3 questions hide right-angled triangles in real-life situations such as a ladder leaning against a wall, a plane flying a direct route, or finding the diagonal of a rectangle. Draw a labelled diagram first, then identify the hypotenuse and the two legs.
许多 KS3 题目将直角三角形隐藏在现实情境中，例如梯子靠在墙上、飞机直飞航线或求矩形的对角线。先画出带标注的示意图，然后确定斜边和两条直角边。
th 5 m is placed with its foot 3 m from a wall. How high up the wall does the ladder reach? Let the height be h. Then h² = 5² − 3² = 25 − 9 = 16, so h = 4 m.
例：一架 5 米长的梯子，梯脚距墙 3 米。梯子能到达墙上的多高处？设高度为 h，则 h² = 5² − 3² = 25 − 9 = 16，因此 h = 4 米。
8. Pythagoras on coordinate grids | 坐标系中的勾股定理
You may be given two points on a coordinate grid and asked to find the distance between them. Draw a right-angled triangle with the horizontal and vertical differences as the two shorter sides. The distance between the points is the hypotenuse.
考试中可能给出坐标系中的两个点，要求你求它们之间的距离。画出以水平差和竖直差为两条直角边的直角三角形。两点之间的距离就是斜边。
and the vertical difference is 6 − 2 = 4. Distance² = 3² + 4² = 9 + 16 = 25, so distance = √25 = 5 units.
例：求点 (1, 2) 和点 (4, 6) 之间的距离。水平差为 4 − 1 = 3，竖直差为 6 − 2 = 4。距离² = 3² + 4² = 9 + 16 = 25，因此距离 = √25 = 5 个单位。
This method is especially useful when the points are not on the same horizontal or vertical line.
当两个点不在同一条水平线或竖直线上时，这个方法尤其有用。
9. Common mistakes to avoid | 常见错误与避免方法
A common error is adding the squares when finding a shorter side. Remember: hypotenuse squared minus known shorter side squared gives the missing shorter side squared. Another error is forgetting to take the square root at the end.
一个常见错误是在求较短直角边时仍然把平方相加。要记住：斜边的平方减去已知直角边的平方，才能得到未知直角边的平方。另一个错误是最后忘记开平方。
Also avoid rounding too early in multi-step calculations. Keep the exact square root value until the final step, then round to the required degree of accuracy, such as 1 decimal place or 3 significant figures.
还要避免在多步计算中过早四舍五入。保留精确的平方根值直到最后一步，再按要求的精确度取近似值，例如 1 位小数或 3 位有效数字。
Finally, check that the hypotenuse is the longest side. If your calculated hypotenuse is shorter than one of the legs, you have probably mixed up the sides.
最后，检查斜边是否为最长边。如果你算出的斜边比某条直角边还短，很可能是把边搞混了。
10. Worked exam-style example | 考试题型示例
Question: In rectangle PQRS, PQ = 12 cm and QR = 9 cm. Find the length of the diagonal PR.
题目：在矩形 PQRS 中，PQ = 12 厘米，QR = 9 厘米。求对角线 PR 的长度。
and PR is the hypotenuse. So PR² = 12² + 9² = 144 + 81 = 225. Therefore PR = √225 = 15 cm.
解：对角线 PR 把矩形分成两个直角三角形。在三角形 PQR 中，PQ 和 QR 是直角边，PR 是斜边。因此 PR² = 12² + 9² = 144 + 81 = 225。所以 PR = √225 = 15 厘米。
This type of question often appears without a diagram, so drawing your own labelled rectangle is essential.
这类题目经常不给出图形，因此自己画出带标注的矩形非常重要。
11. Practice and revision tips | 练习与复习建议
e roots such as √144 = 12 and √169 = 13. Use Pythagorean triples to check answers quickly.
练习计算器和非计算器混合题型。对于非计算器题目，至少熟悉到 15² 的平方数以及常见平方根，例如 √144 = 12 和 √169 = 13。使用勾股数快速检查答案。
When revising, write a checklist: identify the right angle, label the hypotenuse, choose the correct form of the formula, substitute carefully, then solve. Check that your final side length is positive and shorter than the hypotenuse.
复习时，写一份检查清单：确定直角、标出斜边、选择正确的公式形式、仔细代入、然后求解。检查最终边长是否为正且小于斜边。
With regular practice, you
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