📚 Mastering Pythagoras' Theorem for KS3 Cambridge Mathematics | 掌握剑桥 KS3 数学毕达哥拉斯定理
Welcome to this KS3 Cambridge Mathematics revision article on Pythagoras' Theorem. This topic appears frequently in Cambridge Checkpoint tests and lays the foundation for trigonometry and vector work in later stages. Understanding how to apply the theorem to right-angled triangles will help you solve problems involving lengths, distances and real-life measurements.
欢迎阅读这篇 KS3 剑桥数学关于毕达哥拉斯定理的复习文章。该主题经常出现在剑桥 Checkpoint 考试中，并为后续的三角学和向量学习奠定基础。理解如何在直角三角形中应用该定理，将帮助你解决涉及长度、距离和实际测量的问题。
1. What is Pythagoras' Theorem? | 什么是毕达哥拉斯定理？
Pythagoras' Theorem states that in any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
毕达哥拉斯定理指出：在任何一个直角三角形中，斜边长度的平方等于另外两条直角边长度的平方和。
The theorem only works for right-angled triangles, so always check that the triangle has a 90° angle before using it.
该定理只适用于直角三角形，因此在使用之前一定要确认三角形中有一个 90° 角。
a² + b² = c²
Here c represents the hypotenuse, while a and b represent the two shorter sides called the legs.
这里 c 代表斜边，a 和 b 代表两条较短的边，称为直角边。
2. Identifying the Hypotenuse | 识别斜边
The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.
斜边是直角三角形中最长的边，并且始终位于直角（90°角）的对面。
Do not assume the hypotenuse is the sloping side in a diagram; it is defined by its position opposite the right angle.
不要假设斜边一定是图中倾斜的那条边；斜边是由它与直角相对的位置决定的。
Labelling the sides correctly as a, b and c is the first step in solving any Pythagoras problem.
正确地将三条边标记为 a、b 和 c 是解决任何毕达哥拉斯定理问题的第一步。
3. The Standard Formula | 标准公式
be written as a² + b² = c², where c is the hypotenuse and a, b are the two shorter sides.
公式可以写成 a² + b² = c²，其中 c 是斜边，a 和 b 是两条较短的直角边。
To find the hypotenuse, add the squares of the two shorter sides and then take the square root.
要求斜边，先将两条直角边的平方相加，然后再开平方。
c = √a² + b²
To find a shorter side, rearrange the formula to a² = c² − b² or b² = c² − a².
要求一条直角边，可将公式变形为 a² = c² − b² 或 b² = c² − a²。
This rearrangement is essential because subtracting the squares gives the missing square of the leg.
这种变形至关重要，因为减去平方后就能得到缺失直角边的平方。
4. Finding the Hypotenuse | 求斜边
shorter sides of 6 cm and 8 cm. To find the hypotenuse, calculate 6² + 8² = 36 + 64 = 100.
假设一个直角三角形的两条直角边分别为 6 厘米和 8 厘米。要求斜边，先计算 6² + 8² = 36 + 64 = 100。
Then take the square root: √100 = 10 cm. The hypotenuse is 10 cm long.
然后开平方：√100 = 10 厘米。斜边的长度为 10 厘米。
Always include the correct unit in your final answer, as missing units can lose marks in a Cambridge Checkpoint test.
最终答案中一定要写上正确的单位，因为在剑桥 Checkpoint 考试中漏写单位会失分。
ed example: For sides of 5 m and 12 m, c² = 5² + 12² = 25 + 144 = 169, so c = √169 = 13 m.
例题：若两条直角边分别为 5 米和 12 米，则 c² = 5² + 12² = 25 + 144 = 169，所以 c = √169 = 13 米。
5. Finding a Shorter Side | 求直角边
subtraction. For example, if c = 13 cm and a = 5 cm, then b² = 13² − 5² = 169 − 25 = 144.
当已知斜边和一条直角边时，使用减法。例如，若 c = 13 厘米，a = 5 厘米，则 b² = 13² − 5² = 169 − 25 = 144。
Therefore b = √144 = 12 cm. This shows that the missing shorter side is 12 cm.
因此 b = √144 = 12 厘米。这表明缺失的直角边长为 12 厘米。
Many students accidentally add the two known squares even when finding a shorter side, so always decide first whether you are looking for the hypotenuse or a leg.
许多学生在求直角边时也会错误地把两个已知数的平方相加，所以在计算前一定要先判断要求的是斜边还是直角边。
A useful check is that the hypotenuse must be the longest side. If your answer for a shorter side comes out larger than the hypotenuse, you have made an error.
一个有效的检查方法是：斜边一定是最长的边。如果求出的直角边比斜边还长，就说明计算出错了。
6. Real-Life Applications | 实际应用
Pythagoras' Theorem is used in many practical situations, such as finding the length of a ladder needed to reach a window, calculating the diagonal of a TV screen, or measuring the distance across a park.
毕达哥拉斯定理在许多实际场景中都有应用，例如求够到窗户所需的梯子长度、计算电视屏幕的对角线，或测量公园两点之间的距离。
In ladder problems, the ladder is usually the hypotenuse, and the ground and wall form the two shorter sides of a right-angled triangle.
在梯子问题中，梯子通常作为斜边，地面和墙壁构成直角三角形的两条直角边。
For example, if a ladder is placed 3 m from a wall and reaches a height of 4 m, the length of the ladder is √3² + 4² = √25 = 5 m.
例如，若一把梯子底端离墙 3 米，顶端达到 4 米高，则梯子的长度为 √3² + 4² = √25 = 5 米。
Another common example is finding the diagonal distance across a rectangular field of length 24 m and width 7 m: √24² + 7² = √576 + 49 = √625 = 25 m.
另一个常见例子是求一个长 24 米、宽 7 米的矩形场地对角线的距离：√24² + 7² = √576 + 49 = √625 = 25 米。
7. Distance on a Coordinate Grid | 坐标网格上的距离
Pythagoras' Theorem can be used to find the distance between two points on a coordinate grid. The horizontal and vertical differences form the two shorter sides.
毕达哥拉斯定理可用于求坐标网格上两点之间的距离。水平差和垂直差构成两条直角边。
s (4, 6), the horizontal difference is 4 − 1 = 3 and the vertical difference is 6 − 2 = 4.
如果点 P 为 (1, 2)，点 Q 为 (4, 6)，则水平差为 4 − 1 = 3，垂直差为 6 − 2 = 4。
The distance PQ is then √3² + 4² = √9 + 16 = √25 = 5 units.
那么距离 PQ 为 √3² + 4² = √9 + 16 = √25 = 5 个单位。
This method works for any two points because the horizontal and vertical distances always meet at a right angle.
这种方法适用于任意两点，因为水平距离和垂直距离总是以直角相交。
In coordinate problems, always subtract the x-coordinates and y-coordinates separately before squaring them.
在坐标问题中，一定要分别将 x 坐标和 y 坐标相减，然后再平方。
8. Pythagoras in 3D | 三维空间中的毕达哥拉斯定理
In three-dimensional problems
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图