📚 Pythagoras’ Theorem: Finding Missing Lengths in Right-Angled Triangles | 勾股定理：求直角三角形中的未知边长
In KS3 Cambridge Mathematics, Pythagoras’ theorem is one of the most important geometric tools for working with right-angled triangles. It links the three sides of a right-angled triangle and allows you to calculate an unknown side when the other two sides are known.
在 KS3 剑桥数学中，勾股定理是研究直角三角形最重要的几何工具之一。它建立了直角三角形三条边之间的关系，使你在已知两条边的情况下计算未知边。
1. Introduction to Pythagoras’ Theorem | 勾股定理简介
A right-angled triangle has one angle of exactly 90°. The side opposite this right angle is called the hypotenuse, and it is always the longest side. The other two sides are called the legs or shorter sides.
直角三角形有一个恰好为 90° 的角。直角所对的边称为斜边，它总是最长的边。另外两条边称为直角边或短边。
Pythagoras’ theorem states that the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two sides. This geometric idea can be turned into a simple algebraic rule.
勾股定理指出，以斜边为边长的正方形面积等于以两条直角边为边长的正方形面积之和。这个几何概念可以转化为一个简单的代数规则。
2. The Formula a² + b² = c² | 公式 a² + b² = c²
If the two shorter sides have lengths a and b, and the hypotenuse has length c, then the theorem can be written as:
如果两条直角边的长度分别为 a 和 b，斜边长度为 c，那么定理可以写作：
a² + b² = c²
Here a and b can be swapped, but c must always be the hypotenuse. This formula only works for right-angled triangles. If the triangle does not have a 90° angle, the relationship is not true.
这里 a 和 b 可以互换，但 c 必须始终表示斜边。该公式只适用于直角三角形。如果三角形没有 90° 角，这个关系就不成立。
For example, in the classic 3-4-5 triangle, we have 3² + 4² = 9 + 16 = 25, and 5² = 25. This confirms that the sides obey the rule.
例如，在经典的 3-4-5 三角形中，3² + 4² = 9 + 16 = 25，而 5² = 25。这证实了三条边符合该规则。
3. Identifying the Hypotenuse | 识别斜边
Before using the formula, you must correctly identify the hypotenuse. Look for the side that does not touch the right angle. It is always opposite the 90° angle and is the longest side.
在使用公式之前，你必须正确识别斜边。寻找不与直角接触的边。它总是与 90° 角相对，并且是最长的边。
A common trick is to mark the right angle with a small square. Then the side across from that square is the hypotenuse. If you label the hypotenuse incorrectly, you may end up with an answer that is too short or too long.
一个常用技巧是用小方块标出直角。那个方块对面的边就是斜边。如果错误标注斜边，你可能会得到过短或过长的答案。
4. Finding the Hypotenuse | 求斜边
Suppose a right-angled triangle has shorter sides of 6 cm and 8 cm. To find the hypotenuse c, substitute a = 6 and b = 8 into the formula.
假设一个直角三角形的两条直角边分别为 6 cm 和 8 cm。要求斜边 c，将 a = 6 和 b = 8 代入公式。
c² = 6² + 8²
Then c² = 36 + 64 = 100. Taking the square root gives c = √100 = 10 cm.
然后 c² = 36 + 64 = 100。开平方得到 c = √100 = 10 cm。
Notice that you should not forget the units. If the given sides are in cm, the hypotenuse is also in cm. Always write the unit in the final answer.
注意不要忘记单位。如果已知边以 cm 为单位，斜边也以 cm 为单位。最终答案中一定要写出单位。
5. Finding a Shorter Side | 求直角边
When the hypotenuse and one shorter side are known, you need to rearrange the formula. For example, if c = 13 m and b = 5 m, find a.
当斜边和一条直角边已知时，你需要重新整理公式。例如，如果 c = 13 m，b = 5 m，求 a。
a² = c² − b²
Substitute the values: a² = 13² − 5² = 169 − 25 = 144. Then a = √144 = 12 m.
代入数值：a² = 13² − 5² = 169 − 25 = 144。然后 a = √144 = 12 m。
This is the most common error point: students sometimes add 13² and 5² instead of subtracting. Always subtract when finding a shorter side.
这是最常见的错误点：学生有时会把 13² 和 5² 相加而不是相减。求直角边时一定要用减法。
6. Checking Your Answer | 检验答案
After calculating a side, always check two things. First, the hypotenuse must be longer than either of the shorter sides. Second, substitute all three values back into a² + b² = c² to confirm the equality.
计算边长后，始终检查两点。第一，斜边必须比任意一条直角边长。第二，将所有三个值代回 a² + b² = c²，确认等式成立。
For the previous example, 5² + 12² = 25 + 144 = 169, and 13² = 169, so the answer is correct. This check takes only a few seconds and can catch careless arithmetic mistakes.
对于前面的例子，5² + 12² = 25 + 144 = 169，而 13² = 169，因此答案正确。这项检查只需几秒钟，却能发现粗心的算术错误。
7. The Converse of Pythagoras | 勾股定理的逆定理
The converse of Pythagoras’ theorem helps you test whether a triangle is right-angled. If a triangle has sides a, b and c, and a² + b² = c², then the triangle is right-angled with the right angle opposite side c.
勾股定理的逆定理帮助你判断一个三角形是否为直角三角形。如果一个三角形的三边为 a、b 和 c，且 a² + b² = c²，那么这个三角形是直角三角形，直角对边为 c。
For example, a triangle with sides 8, 15 and 17 is right-angled because 8² + 15² = 64 + 225 = 289, and 17² = 289. Remember that the largest number must be tested as c, not as a or b.
例如，三边为 8、15 和 17 的三角形是直角三角形，因为 8² + 15² = 64 + 225 = 289，而 17² = 289。请记住，最大的数必须作为 c 来检验，而不是作为 a 或 b。
8. Word Problems and Applications | 应用题与实际应用
Pythagoras’ theorem appears in many real-life problems. A ladder leaning against a wall forms a right-angled triangle with the ground. If a 5 m ladder is placed 3 m from the wall, the height reached is √(5² − 3²) = √(25 − 9) = √16 = 4 m.
勾股定理出现在许多实际问题中。靠在墙上的梯子与地面构成直角三角形。如果一个 5 m 长的梯子底部距离墙 3 m，那么达到的高度为 √(5² − 3²) = √(25 − 9) = √16 = 4 m。
Another example is finding the direct distance between two points on a coordinate grid by treating horizontal and vertical changes as the two shorter sides. Always draw a labelled sketch for word problems. It helps you see which length is the hypotenuse and which sides are given.
另一个例子是在坐标网格中求两点之间的直线距离，将水平变化和垂直变化看作两条直角边。对于应用题，始终画出带标注的草图。这能帮助你看清哪条边是斜边，以及已知哪些边。
9. Common Mistakes | 常见错误
- Using a² + b² = c² for non-right-angled triangles | 对非直角三角形使用 a² + b² = c²
- Adding instead of subtracting when finding a shorter side | 求直角边时相加而不是相减
- Forgetting to take the square root at the end | 最后忘记开平方
- Labelling the hypotenuse as a shorter side | 把斜边标成直角边
- Omitting units in the final answer | 最终答案遗漏单位
To avoid these errors, first confirm the triangle is right-angled, mark the hypotenuse clearly, write the rearranged formula before substituting, and always include units in the final answer.
为了避免这些错误，首先要确认三角形是直角三角形，清楚地标出斜边，在代入前写出整理好的公式，并在最终答案中始终包含单位。
10. Exam Tips | 考试技巧
In Cambridge KS3 exams, marks are often awarded for method. Always write the formula first, substitute the values clearly, and show the square root step. Even if you make a small arithmetic slip, you can still gain method marks.
在剑桥 KS3 考试中，方法步骤通常可以得分。始终先写出公式，清晰地代入数值，并展示开平方的步骤。即使出现小的算术失误，你仍然可以获得方法分。
If you are asked to check whether a triangle is right-angled, calculate both sides of the equation separately before comparing them. If an answer involves a square root that is not exact, leave it in surd form or round to the requested number of decimal places.
如果题目要求判断三角形是否为直角三角形，请分别计算等式两边，再进行比较。如果答案涉及不能开尽的平方根，请保留根式形式或按要求四舍五入到指定小数位。
Published by TutorHao | KS3 Cambridge
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