📚 Constructing Triangles with Ruler and Compass | 用直尺与圆规作三角形
In geometry, constructing triangles accurately is a fundamental skill. Using only a straightedge and a compass, you can draw any triangle if you are given three sides (SSS), two sides and the included angle (SAS), or two angles and the included side (ASA). This article explains each method step by step, with clear instructions and practice problems to build your confidence for the Cambridge KS3 mathematics course.
在几何中，准确作三角形是一项基础技能。仅用一把直尺和一个圆规，只要给出三条边（SSS）、两边及其夹角（SAS）或两角及其夹边（ASA），你就能画出任何三角形。本文将逐步解释每一种方法，并配有清晰的说明和练习题，帮助你在剑桥 KS3 数学课程中建立信心。
1. What You Need | 所需工具
All constructions in this article require only three tools: a sharp pencil, a ruler (straightedge) and a pair of compasses. A protractor is not used for measuring angles in pure compass-and-straightedge constructions, but we will sometimes use it to check our work. Make sure your compass holds its radius firmly, and always draw light construction lines so they can be adjusted if needed.
本文中的所有作图只需要三种工具：一支削尖的铅笔、一把直尺和一支圆规。在纯正的圆规直尺作图中，不用量角器来画角，但我们有时会用它来检验结果。确保圆规能牢牢固定半径，并且所有的辅助线都画得较轻，以便需要时进行调整。
Before starting, label points clearly with capital letters such as A, B and C. This helps you follow the construction steps without confusion. Keep your compass point sharp and avoid pressing so hard that you tear the paper.
开始之前，用大写字母（如 A、B、C）清晰地标记点。这能让你在按照步骤作图时不会混淆。圆规的针尖要保持锋利，下笔不要太重，以免划破纸张。
2. Constructing a Triangle Given Three Sides (SSS) | 已知三边作三角形
4 cm. First, use the ruler to draw side AB exactly 6 cm long. Label the endpoints A and B.
当已知三条边的长度时，使用 SSS 作图法。例如，已知 AB = 6 cm，BC = 5 cm，AC = 4 cm，作三角形 ABC。首先，用直尺画出恰好 6 cm 长的边 AB，并标出端点 A 和 B。
Next, set your compass to 5 cm (the length of BC). Place the compass point on B and draw an arc above the line AB. Then set the compass to 4 cm (the length of AC). Place the point on A and draw a second arc that intersects the first. The intersection is point C. Finally, join A to C and B to C with straight lines to complete the triangle.
接下来，将圆规设置到 5 cm（BC 的长度）。把圆规针尖放在点 B 上，在 AB 上方画一条弧。再把圆规设置到 4 cm（AC 的长度），针尖放在点 A 上，画出第二条弧，与第一条弧相交。交点就是点 C。最后，用直线连接 AC 和 BC，三角形就完成了。
Remember, the arcs must cross; if they do not meet, check your compass settings and the lengths. Also, draw the arcs lightly so that unwanted marks can be ignored later.
记住，两条弧必须相交；如果不相交，检查圆规的设置和长度。另外，画弧时要轻，这样多余的痕迹之后可以忽略不计。
3. Constructing a Triangle Given Two Sides and the Included Angle (SAS) | 已知两边及其夹角作三角形
= 7 cm, PR = 5 cm and angle QPR = 50°. Start by drawing side PQ 7 cm long. Label P and Q.
在 SAS 情况下，你已知两条边以及它们之间的夹角（即夹角）。假设需要作三角形 PQR，已知 PQ = 7 cm，PR = 5 cm，∠QPR = 50°。先画出 7 cm 长的边 PQ，标出 P 和 Q。
Place the protractor at point P and measure an angle of 50° from PQ. Draw a faint line along this direction, longer than 5 cm. On this ray, use the ruler to mark point R exactly 5 cm from P. Then simply join R to Q with a straight line, and triangle PQR is formed.
将量角器放在点 P，从 PQ 开始量出一个 50° 的角。沿这个方向画一条轻的射线，长度大于 5 cm。在这条射线上，用直尺从点 P 量出 5 cm 标出点 R。然后用直线连接 R 和 Q，三角形 PQR 就形成了。
If you want to do a pure compass-and-straightedge construction without a protractor, you would need to copy the given angle using the compass. However, at KS3 level, using a protractor for the SAS construction is accepted and often more straightforward.
如果想做纯粹的圆规直尺作图而不使用量角器，那就需要用圆规来复制已知角度。但是在 KS3 阶段，对 SAS 作图使用量角器通常是被接受的，而且更直接。
4. Constructing a Triangle Given Two Angles and the Included Side (ASA) | 已知两角及其夹边作三角形
gle XYZ = 40° and angle YXZ = 60°. Draw base XY exactly 8 cm long and mark points X and Y.
当已知的边位于两个给定角之间时，使用 ASA 方法。作三角形 XYZ，已知 XY = 8 cm，∠XYZ = 40°，∠YXZ = 60°。画出恰好 8 cm 长的底边 XY，标记点 X 和 Y。
At point Y, use the protractor to measure and draw a ray at 40° to XY. At point X, draw a ray at 60° to XY on the same side of the base. The two rays will meet at point Z, which is the third vertex of the triangle. Erase any extra lengths beyond Z and darken the triangle sides.
在点 Y 用，用量角器量出与 XY 成 40° 的角，画出一条射线。在点 X 处，在底边同侧画一条与 XY 成 60° 的射线。两条射线相交于点 Z，这就是三角形的第三个顶点。擦去 Z 点以外的多余线，加深三角形的边。
Make sure the two rays are drawn on the same side of XY, otherwise they might diverge and never meet. A quick angle-sum check reminds you that the third angle at Z must be 80° if the other two are 40° and 60°, since interior angles of a triangle sum to 180°.
确保两条射线画在 XY 的同一侧，否则它们可能向外发散而永远不相交。快速验算一下角度和，如果另两个角是 40° 和 60°，那么第三个角 Z 必定是 80°，因为三角形内角和为 180°。
5. Understanding Included Angle and Included Side | 理解夹角与夹边
One common source of error is confusing the 'included' part. In SAS, the angle must be the one sandwiched between the two known sides. If you are given side AB, side BC and angle A, that is not SAS because the angle is not between the two given sides. Such a configuration may produce two possible triangles, which is a more advanced topic.
一个常见的错误来源是混淆“夹”的含义。在 SAS 中，给定角必须是夹在两条已知边之间的那个角。如果你已知边 AB、边 BC 和角 A，那并不是 SAS，因为这个角并不在两条已知边之间。这样的条件可能产生两个可能的三角形，这是一个更高级的主题。
In ASA, the side must be the one joining the two vertices where the angles are drawn. If you know angle A, angle B and side AC, then the side is not included; again, you risk creating an ambiguous case. Always check that the given side is exactly the side between the two angles.
在 ASA 中，已知边必须是连接两个角顶点的边。如果你知道角 A、角 B 和边 AC，那么边并不是夹边；同样，有可能造成不定情况。务必检查已知边恰好是两角之间的那一条边。
6. Checking Your Construction | 检验你的作图
After completing a triangle construction, always verify your work. Measure the third side with your ruler and compare it to the expected length if it was given. For SSS, check that all three sides match the given lengths. For SAS, measure the remaining side and the other two angles to see if the triangle is consistent.
完成三角形的作图后，一定要检验你的作品。用直尺测量第三条边，与预期长度比较（如果题目给了）。对于 SSS，检查三条边是否都符合给定的长度。对于 SAS，测量剩下的边以及另外两个角，看三角形是否一致。
Use a protractor to check angles, placing the centre on the vertex and aligning one arm with a side. The smallest error in arc intersections can lead to a side being 1-2 mm off, which is usually acceptable in a pencil construction, but aim for precision.
用量角器检查角度，将量角器的中心放在顶点上，一条边对准三角形的边。弧线相交时的微小误差可能导致边长出现 1–2 毫米的偏差，这在铅笔作图中通常是可以接受的，但要力求精确。
7. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Mistake 1: Compass radius slips while drawing arcs. Always tighten the compass screw firmly; if it still moves, hold the compass by the top handle and avoid pressing sideways. Mistake 2: Measuring from the wrong end of the ruler. Always align the zero mark exactly with the starting point.
错误 1：画弧时圆规半径滑动。始终拧紧圆规的螺丝；如果还是会动，握住圆规顶部的手柄，避免侧向按压。错误 2：从直尺错误的一端开始测量。始终将零刻度线精确对准起点。
Mistake 3: Drawing arcs that are too short, so they do not clearly intersect. Draw arcs that sweep at least a quarter-circle in the region where the vertex is expected. Mistake 4: Using a blunt pencil, which gives thick, inaccurate lines. Keep your pencil sharp at all times.
错误 3：弧画得太短，导致无法清晰相交。在预期顶点区域，至少画出一个四分之一圆的弧线。错误 4：铅笔太钝，画出的线条粗且不精确。始终保持铅笔削尖。
8. Real-Life Applications | 实际应用
Triangle constructions are not just academic exercises. Architects and engineers use similar geometric principles when designing roof trusses, bridges and ramps. Understanding how a triangle is uniquely determined by SSS, SAS or ASA helps ensure structural stability, since a triangle is the only rigid polygon.
三角形作图不仅仅是学术练习题。建筑师和工程师在设计屋架、桥梁和坡道时，会用到类似的几何原理。理解三角形如何由 SSS、SAS 或 ASA 唯一确定，有助于确保结构稳定性，因为三角形是唯一具有刚性的多边形。
In navigation and map-making, constructing triangles from known distances and bearings relies on the same SAS and ASA ideas. Even simple tasks like plotting a garden layout or checking a picture frame for squareness can involve constructing a right triangle using the 3-4-5 SSS rule.
在导航和地图绘制中，从已知距离和方位角作三角形，也依赖于相同的 SAS 和 ASA 思想。即便是测量花园布局或检查相框是否方正这样简单的任务，也可能用到 3-4-5 的 SSS 规则来构造直角三角形。
9. Practice Problem Walkthrough (SSS) | 练习题详解（SSS）
olution: First, draw base DE exactly 5.5 cm. Set compass to 6.5 cm, place at E, and draw an arc upwards. Set compass to 4.5 cm, place at D, and draw an intersecting arc. Label the intersection F. Join DF and EF to finish.
问题：已知 DE = 5.5 cm，EF = 6.5 cm，FD = 4.5 cm，作三角形 DEF。解答：首先，精确画出 5.5 cm 长的底边 DE。将圆规设为 6.5 cm，针尖放在 E 点，向上方画弧。再将圆规设为 4.5 cm，针尖放在 D 点，画另一条弧与之相交。标出交点 F。连接 DF 和 EF 即完成。
After construction, measure side FD: it should be 4.5 cm. Measure angles D and E to verify that the shape matches the given data. Practice this a few times to get consistent arcs.
作出三角形后，测量边 FD，应为 4.5 cm。测量角 D 和角 E，验证图形是否与给定数据一致。多次练习以便画出稳定的弧线。
10. Practice Problem Walkthrough (SAS) | 练习题详解（SAS）
Solution: Draw GH 8 cm. Place protractor at H, measure 35° from GH, and draw a long ray. On this ray, measure 6 cm from H and mark point I. Connect G to I. The construction is complete.
问题：已知 GH = 8 cm，HI = 6 cm，∠GHI = 35°，作三角形 GHI。解答：画出 8 cm 长的 GH。将量角器放在 H 点，从 GH 起量出 35°，画一条较长的射线。在这条射线上，从 H 量出 6 cm 标出点 I。连接 GI，作图完成。
Check side GI with a ruler; its length should be around 4.9 cm to 5.1 cm depending on accuracy. The angle at I should measure approximately 105°, and angle G about 40°.
用直尺检查边 GI；根据精确度，其长度应在 4.9 cm 到 5.1 cm 左右。角 I 应该约为 105°，角 G 约为 40°。
11. Practice Problem Walkthrough (ASA) | 练习题详解（ASA）
, draw a ray at 45° to JK. At K, draw a ray at 65° to JK on the same side. The intersection is L. Complete the triangle by joining J, K and L.
问题：已知 JK = 7 cm，∠JKL = 65°，∠KJL = 45°，作三角形 JKL。解答：画出 7 cm 长的边 JK。在 J 点画一条与 JK 成 45° 的射线。在 K 点，在同侧画一条与 JK 成 65° 的射线。两条射线的交点为 L。连接 J、K、L 即完成三角形。
otractor to confirm. If the angle differs by more than 2°, redraw the rays more carefully.
第三个角 L 应为 70°，因为 180° - 65° - 45° = 70°。用量角器验证。若角度偏差超过 2°，需要更仔细地重新画射线。
12. Summary | 总结
Constructing triangles with ruler and compass builds spatial reasoning and precision. SSS requires two intersecting arcs; SAS needs an accurately measured angle; ASA needs two rays from the endpoints of a side. Always label points, draw fine lines, and check with your ruler and protractor. These skills are essential for the Cambridge KS3 curriculum and will be used again in higher-level geometry.
用直尺和圆规作三角形可以培养空间推理能力和精确度。SSS 需要两条相交的弧；SAS 需要精确测量角度；ASA 需要从边的两端画两条射线。始终要标记点、画细线，并用直尺和量角器检查。这些技能是剑桥 KS3 课程的核心内容，在以后更高层次的几何学习中还会反复用到。
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