📚 Pythagoras' Theorem and Right-Angled Triangle Problems | 勾股定理与直角三角形应用
In KS3 Cambridge Mathematics, Pythagoras' theorem and basic right-angled triangle trigonometry are essential tools for solving problems involving lengths and angles. Understanding these ideas helps you break down complex diagrams into right-angled triangles and apply reliable methods.
在 KS3 剑桥数学中，勾股定理和基本的直角三角形三角比是解决涉及长度和角度问题的重要工具。理解这些概念能帮助你将复杂图形分解为直角三角形，并应用可靠的方法。
1. The Pythagorean Equation | 勾股定理方程
Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is the side opposite the right angle and is always the longest side.
勾股定理指出，在任何直角三角形中，斜边的平方等于另外两条直角边的平方和。斜边是直角所对的边，并且总是最长的边。
a² + b² = c²
Here a and b represent the two shorter legs, while c represents the hypotenuse. You can rearrange this equation to find any missing side when the other two sides are known.
这里 a 和 b 表示两条较短的直角边，c 表示斜边。当已知另外两条边时，你可以重新整理这个方程来求任意缺失的边。
2. Identifying the Hypotenuse and Legs | 识别斜边与直角边
Before substituting values, always identify the hypotenuse first. The hypotenuse is opposite the right angle and is never one of the sides that form the right angle.
在代入数值之前，总是先识别斜边。斜边与直角相对，绝不是构成直角的两条边之一。
A common error is to label the longest side as a or b rather than c. Although the theorem works if you keep the labels consistent, using c for the hypotenuse reduces mistakes.
一个常见的错误是把最长的边标为 a 或 b 而不是 c。虽然保持标签一致时定理仍然成立，但用 c 表示斜边可以减少错误。
In a right-angled triangle, the right angle is often marked with a small square. The side directly across from that square is the hypotenuse, and the other two sides are the legs.
在直角三角形中，直角通常用一个小方框标出。与该方框直接相对的边就是斜边，另外两条边是直角边。
3. Finding the Hypotenuse | 求斜边
example, if a = 6 cm and b = 8 cm, then c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
求斜边时，将两条较短边代入方程并解出 c。例如，如果 a = 6 cm，b = 8 cm，那么 c² = 6² + 8² = 36 + 64 = 100，因此 c = √100 = 10 cm。
c = √a² + b²
Remember to take the positive square root because a length cannot be negative. In a real problem, give the answer with the correct units and round only if the question asks you to.
请记住取正平方根，因为长度不能为负数。在实际问题中，需要给出带正确单位的答案，并且只有题目要求时才进行四舍五入。
= 5² + 12² = 25 + 144 = 169, so c = 13 m. This is a useful Pythagorean triple to remember.
再举一个例子，如果 a = 5 m，b = 12 m，那么 c² = 5² + 12² = 25 + 144 = 169，因此 c = 13 m。这是一组值得记住的有用勾股数。
4. Finding a Shorter Side | 求直角边
he other leg. If c = 13 cm and a = 5 cm, then b² = c² - a² = 169 - 25 = 144, so b = 12 cm.
当已知斜边和一条直角边时，重新整理方程求出另一条直角边。如果 c = 13 cm，a = 5 cm，那么 b² = c² - a² = 169 - 25 = 144，因此 b = 12 cm。
a = √c² - b²
This rearrangement is useful for problems such as finding the height of a ladder against a wall when the ladder length and ground distance are given.
这种重新整理在类似求梯子靠墙的高度问题中很有用，题目通常会给出梯子长度和地面距离。
Another example: if the hypotenuse is 17 cm and one leg is 8 cm, then the missing leg is √17² - 8² = √289 - 64 = √225 = 15 cm.
另一个例子：如果斜边为 17 cm，一条直角边为 8 cm，那么缺失的直角边为 √17² - 8² = √289 - 64 = √225 = 15 cm。
5. Pythagorean Triples | 勾股数
these are called Pythagorean triples. The simplest example is 3-4-5 because 3² + 4² = 5².
一些整数边长恰好满足该定理；这些数被称为勾股数。最简单的例子是 3-4-5，因为 3² + 4² = 5²。
Recognising triples such as 5-12-13 and 8-15-17 can speed up your work. However, you must check which side is the hypotenuse before matching a triple.
识别 5-12-13 和 8-15-17 等勾股数可以加快解题速度。但你必须先确认哪条边是斜边，再匹配勾股数。
| Leg a | 直角边 a | Leg b | 直角边 b | Hypotenuse c | 斜边 c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
Memorising the common triples helps you spot missing sides without calculating every square root. Multiples of a triple also work, such as 6-8-10 and 9-12-15.
记住常见的勾股数有助于你在不必计算每个平方根的情况下发现缺失的边。勾股数的倍数也成立，例如 6-8-10 和 9-12-15。
6. Using Pythagoras in Composite Shapes | 复合图形中的勾股定理
Many exam questions ask you to use Pythagoras' theorem after drawing a right-angled triangle inside a rectangle, square, or trapezium. Sketch the diagonal or height and label all known lengths.
许多考试题目会要求你在矩形、正方形或梯形内部画出直角三角形后使用勾股定理。画出对角线或高，并标出所有已知长度。
For example, in a rectangle with width 12 m and height 5 m, the diagonal is the hypotenuse of a right-angled triangle, so its length is √12² + 5² = 13 m.
例如，在一个宽 12 m、高 5 m 的矩形中，对角线是直角三角形的斜边，因此其长度为 √12² + 5² = 13 m。
In an isosceles triangle, the perpendicular height from the apex to the base bisects the base. You can form two identical right-angled triangles and use Pythagoras to find the height or the equal side length.
在等腰三角形中，从顶点到底边的垂直高平分底边。你可以形成两个全等的直角三角形，并使用勾股定理求出高或相等的边长。
For a square with side 10 cm, the diagonal is √10² + 10² Always draw the right-angled triangle you are using before substituting into the theorem.
对于边长为 10 cm 的正方形，其对角线为 √10² + 10² = √200 ≈ 14.1 cm。在代入定理之前，始终先画出你正在使用的直角三角形。
7. Right-Angled Triangle Trigonometry: Sine, Cosine, Tangent | 直角三角形三角比：正弦、余弦、正切
In a right-angled triangle, the three basic trigonometric ratios relate an acute angle to two sides. For an angle θ, the ratios are defined as follows.
在直角三角形中，三个基本三角比将一个锐角与两条边联系起来。对于角 θ，这些比定义如下。
| Ratio | 比 | Formula | 公式 |
|---|---|
| Sine | 正弦 | sin θ = opposite / hypotenuse | sin θ = 对边 / 斜边 |
| Cosine | 余弦 | cos θ = adjacent / hypotenuse | cos θ = 邻边 / 斜边 |
| Tangent | 正切 | tan θ = opposite / adjacent | tan θ = 对边 / 邻边 |
The opposite side is opposite the angle θ, the adjacent side is next to θ but is not the hypotenuse, and the hypotenuse remains the longest side.
对边是与角 θ 相对的边，邻边是与 θ 相邻但不是斜边的边，斜边仍然是最长的边。
OA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
一个有用的记忆口诀是 SOH CAH TOA：正弦 = 对边/斜边，余弦 = 邻边/斜边，正切 = 对边/邻边。
8. Finding Missing Sides with Trigonometry | 用三角比求未知边
if you know the hypotenuse and need the opposite side, use sin θ = opposite / hypotenuse.
要找到未知边，选择包含已知边和所求边的三角比。例如，如果你知道斜边而需要求对边，就使用 sin θ =
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