Constructing Triangles Using a Ruler and Compass | 用直尺和圆规作三角形

📚 Constructing Triangles Using a Ruler and Compass | 用直尺和圆规作三角形

In KS3 Cambridge Mathematics, constructing accurate triangles is a fundamental skill that combines geometry, measurement and logical thinking. Instead of just sketching, you learn to use a ruler, compass and protractor to draw triangles that exactly match given conditions. These constructions are based on triangle congruence rules – SSS, SAS, ASA and RHS – and they form the foundation for more advanced topics like scale drawings, loci and even Pythagoras’ theorem. This article will guide you through the key methods, step-by-step examples and common pitfalls so you can confidently tackle any triangle construction question on your homework or the Checkpoint test.

在 KS3 剑桥数学中，精确作三角形是一项融合几何、测量与逻辑思维的基本技能。你不仅要画草图，还要学会使用直尺、圆规和量角器，画出完全符合给定条件的三角形。这些作图方法基于三角形的全等判定法则——SSS、SAS、ASA 和 RHS，它们为后续的缩放图、轨迹甚至勾股定理等高级主题打下基础。本文将带你掌握主要的作图方法、分步实例以及常见错误，让你能够自信地应对作业或 Checkpoint 考试中任何三角形作图题。

---

1. Introduction to Triangle Construction | 三角形作图简介

In geometry, a triangle is uniquely defined if you know certain combinations of its sides and angles. These combinations correspond to the four congruence rules. When you construct a triangle using a ruler and compass, you are actually applying these rules to physically recreate the shape. The process teaches you to be precise, to plan a sequence of arcs and lines, and to leave your construction marks visible for checking. On page 113 of your Cambridge Maths workbook, you will often find exercises that ask you to construct triangles with given measurements and then measure missing parts to verify accuracy.

在几何中，如果知道一个三角形的某些边和角的组合，它的形状和大小就被唯一确定了。这些组合正好对应四种全等判定法则。当你用直尺和圆规作三角形时，实际上就是在实际应用这些法则去重建图形。这个过程训练你精确作图、规划弧线和直线的顺序，并保留作图痕迹以便检查。在你的剑桥数学练习册第 113 页，你经常会看到要求根据给定尺寸作三角形、然后测量缺失部分来验证准确度的习题。

---

2. Tools Needed | 所需工具

For accurate triangle constructions, you will need a sharp pencil, a ruler with centimetre and millimetre markings, a pair of compasses, a protractor and an eraser. The pencil must be kept sharp so that arcs and lines are thin and clear. The compass should be adjusted carefully so its radius does not slip while drawing arcs. A protractor is used when angles are given, but many constructions rely primarily on the compass to copy lengths. Always check that your ruler is straight and that your compass has a firm hinge.

为了精确作三角形，你需要一支削尖的铅笔、一把带厘米和毫米刻度的直尺、一副圆规、一个量角器和一块橡皮。铅笔必须保持尖锐，这样画出的弧线和直线才细而清晰。调节圆规时要小心，以免画弧时半径发生滑动。当给定角度时需要使用量角器，但在很多作图中主要依靠圆规来复制长度。一定要检查直尺是否平直，圆规的铰链是否紧实。

---

3. Constructing a Triangle Given Three Sides (SSS) | 已知三边作三角形 (SSS)

The SSS construction is the most straightforward: if you know the lengths of all three sides, you can draw the triangle using only a ruler and compass. Start by drawing the longest side as a base line segment. Then, set your compass to the length of the second side and draw an arc from one endpoint of the base. Set the compass to the third side and draw an arc from the other endpoint. The point where the arcs intersect is the third vertex. Finally, connect this vertex to both endpoints to complete the triangle. This method works because the third vertex is uniquely located at the intersection of the two arcs.

SSS 作图是最直接的一种：如果已知三条边的长度，只需直尺和圆规就能画出三角形。先画出最长的一边作为底边线段。然后将圆规调到第二条边的长度，从底边的一个端点画一条弧；再将圆规调到第三条边的长度，从另一个端点画弧。两条弧的交点就是第三个顶点的位置。最后，将这个顶点与底边的两个端点分别连接，三角形就完成了。这个方法可行是因为第三个顶点唯一地落在两条弧的交点上。

---

4. Constructing a Triangle Given Two Sides and the Included Angle (SAS) | 已知两边及其夹角作三角形 (SAS)

In an SAS construction, you are given two sides and the angle between them. Draw one of the given sides as a base. At one end, use a protractor to measure and mark the given angle, then draw a long ray along that direction. Set your compass to the length of the second side, place the point at the same endpoint, and draw an arc that cuts the ray. The intersection is the third vertex. Join it to the other endpoint of the base. The protractor is essential here to set the correct angle, but the compass ensures the second side is exactly the right length.

在 SAS 作图中，已知两边及其夹角。先画出其中一条边作为底边。在底边的一个端点处，用量角器量出给定的角度并做标记，然后沿这个方向画一条较长的射线。把圆规调到第二条边的长度，针尖放在同一个端点，画弧与射线相交。交点就是第三个顶点。再把它与底边的另一个端点连接起来。这里量角器是设置正确角度的关键，而圆规则保证了第二条边的长度精确无误。

---

5. Constructing a Triangle Given Two Angles and the Included Side (ASA) | 已知两角及其夹边作三角形 (ASA)

When two angles and the side between them are known, start by drawing the given side. At one endpoint, use a protractor to draw a ray at the first angle. At the other endpoint, draw a ray at the second angle on the same side of the line segment. The two rays will meet at the third vertex. No compass is needed for side lengths, but you may use a compass to check equal angles if required. This construction is used often in scale drawings and bearing problems.

当已知两个角和它们之间的夹边时，先画出给定的那条边。在其一个端点处，用量角器画出第一个角所在的射线；在另一个端点处，在线段的同一侧画出第二个角所在的射线。两条射线的交点就是第三个顶点。这里不需要用圆规量取边长，但有时可以用圆规来检查角度是否相等。这一作图方法在缩放图和方位角问题中很常用。

---

6. Constructing a Right-Angled Triangle Given the Hypotenuse and One Side (RHS) | 已知斜边和一条直角边作直角三角形 (RHS)

The RHS construction creates a right-angled triangle. First, draw the given shorter side (one leg) as a base. At one endpoint, construct a perpendicular line using a compass and straightedge or a protractor. Set your compass to the length of the hypotenuse, place the point at the other endpoint of the base, and draw an arc that cuts the perpendicular line. The intersection is the third vertex. Join it to the endpoints. The right angle ensures the triangle is uniquely determined by the hypotenuse and one leg.

RHS 作图可以画出直角三角形。先画出给定的一条直角边作为底边。在它的一个端点处，用圆规和直尺或量角器作一条垂线。将圆规调到斜边的长度，针尖放在底边的另一个端点处，画弧与垂线相交。交点就是第三个顶点。把它与两个端点连接起来。直角条件确保了由斜边和一条直角边可以唯一确定三角形。

---

7. Step-by-Step Example: SSS Construction | SSS 作图实例

Let us construct triangle ABC with AB = 5 cm, BC = 4 cm and CA = 6 cm. Step 1: draw the longest side, CA = 6 cm, as a horizontal base. Label endpoints C and A. Step 2: set compass to 5 cm (length AB). Place the point at A and draw a large arc above the base. Step 3: set compass to 4 cm (length BC). Place the point at C and draw a second arc crossing the first. Label the intersection B. Step 4: use a ruler to draw line segments AB and BC. Check that AB is indeed 5 cm and BC is 4 cm. You have now constructed the triangle accurately.

让我们作一个三角形 ABC，其中 AB = 5 cm，BC = 4 cm，CA = 6 cm。第 1 步：画出最长边 CA = 6 cm 作为水平底边。标出端点 C 和 A。第 2 步：将圆规调到 5 cm（AB 的长度），针尖放在 A 点，在底边上方画一条大弧。第 3 步：将圆规调到 4 cm（BC 的长度），针尖放在 C 点，画出第二条弧与第一条弧相交。将交点标为 B。第 4 步：用直尺画出线段 AB 和 BC。检查 AB 是否真的是 5 cm，BC 是否为 4 cm。现在你已经准确地作出了这个三角形。

---

8. Step-by-Step Example: SAS Construction | SAS 作图实例

Construct triangle PQR with PQ = 7 cm, PR = 5 cm and ∠P = 40°. Step 1: draw base PQ = 7 cm. Step 2: place the protractor centre at P, align the baseline with PQ, and mark 40°. Draw a ray from P through the mark. Step 3: set compass to 5 cm (PR), place the point at P, and draw an arc cutting the ray. Label the intersection R. Step 4: join R to Q. Measure angle Q and side QR to see if they match expectations – but the construction is complete once R is found.

作三角形 PQR，已知 PQ = 7 cm，PR = 5 cm，∠P = 40°。第 1 步：画出底边 PQ = 7 cm。第 2 步：将量角器中心放在 P 点，底线与 PQ 对齐，标记 40°。从 P 点出发穿过标记画一条射线。第 3 步：把圆规调到 5 cm（PR），针尖放在 P 点，画弧与射线相交。交点标为 R。第 4 步：连接 R 和 Q。你可以测量 ∠Q 和边 QR 看是否符合预期——但一旦找到 R 点，作图就完成了。

---

9. Common Mistakes and Tips | 常见错误与技巧

One frequent error is not leaving construction arcs visible. In exams, you must show your compass arcs to prove you used the correct method. Another mistake is letting the compass slip while drawing, which changes the radius. Always tighten the compass screw properly and handle it gently. Some students confuse the order of SAS: the given angle must be between the two given sides. For ASA, make sure both angles are drawn on the same side of the base. Finally, always use a sharp pencil and draw thin lines – thick lines reduce accuracy and make intersections unclear.

一个常见错误是没有保留作图弧线。考试中，你必须展示圆规弧线以证明使用了正确的方法。另一个错误是画弧时圆规滑动，导致半径变化。一定要拧紧圆规螺丝并轻拿轻放。有些学生容易搞混 SAS 的条件：给定的角必须位于两条已知边之间。对于 ASA，要确保两个角都画在底边的同一侧。最后，始终使用削尖的铅笔画细线——粗线会降低精度，使交点模糊不清。

---

10. Accuracy in Construction | 作图的精确性

To achieve high accuracy, hold the compass at the top and rotate it smoothly without pressing too hard. Draw a long enough arc so that the intersection is clearly defined. When using a protractor, read the scale carefully from the correct direction (inner or outer scale) depending on where your angle starts. Always double-check the length of each side after construction. Small errors in measuring angles or setting the compass radius can lead to a triangle that does not quite close, which tells you to repeat the construction more carefully.

为了实现高精度，拿圆规时要握住顶部，平稳旋转，不要用力过猛。画出的弧线要足够长，以便交点清晰可见。使用量角器时，要根据角度的起始方向，仔细辨认该读内圈还是外圈的刻度。作完图后，务必再次检查每条边的长度。测量角度或调节圆规半径时的微小误差，可能导致三角形无法完全闭合，这就提示你需要更仔细地重新作图。

---

11. Real-World Applications | 实际应用

Triangle constructions are not just classroom exercises. Surveyors use similar techniques to map land and determine distances. Engineers apply triangle principles when designing bridges, roof trusses and mechanical parts. In navigation, constructing a triangle can help find positions from bearings. Understanding these constructions also strengthens your grasp of congruence, similarity and the basis of geometric proofs. The patience and precision you develop here are skills that benefit many areas of mathematics and science.

三角形作图并不仅仅是课堂练习。测量员使用类似的技术绘制地图和计算距离。工程师在设计桥梁、屋架和机械零件时会应用三角形原理。在航海中，作三角形有助于通过方位角确定位置。理解这些作图方法还能巩固你对全等、相似以及几何证明基础的认识。你在这里培养的耐心和精确度，是对许多数学和科学领域都有益处的技能。

---

12. Summary | 总结

To construct triangles accurately, you must identify which congruence rule applies – SSS, SAS, ASA or RHS – and then follow a clear sequence of drawing baseline, arcs and measurements. Always use sharp instruments, show all construction arcs, and check your work. Page 113 of your Cambridge Maths workbook provides excellent practice in applying these methods. Mastering these constructions will give you a strong foundation for more advanced geometry and for the Checkpoint test.

要准确地作出三角形，你必须先判断适用哪种全等判定法则——SSS、SAS、ASA 或 RHS——然后按照清晰的顺序画出底边、弧线和测量标记。始终使用尖锐的工具，保留所有作图弧线，并检查你的成果。剑桥数学练习册的第 113 页为应用这些方法提供了极好的练习。掌握这些作图方法将为你学习更高阶的几何和为 Checkpoint 考试打下坚实的基础。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）