Did you know? You can find the exact middle of a line segment using a compass and a straightedge, even without measuring half its length with a ruler first. The marks made by equal-radius arcs reveal why the construction works. That is the fascinating part of geometry: a drawing can be a mathematical explanation.
Quick answer for Bukit Timah parents: the 2027 SEC G1 Mathematics K110 syllabus includes properties and constructions of perpendicular bisectors and angle bisectors. Students should know the difference between these two lines, how to make the construction with instruments, and how to verify their result.
What does a perpendicular bisector do?
Imagine a straight line segment AB measuring 8 cm. Its midpoint M is 4 cm from A and 4 cm from B. A perpendicular bisector is a line that passes through M and meets AB at 90°. It both bisects the segment and is perpendicular to it.
The important feature is not whether the drawing looks vertical or horizontal. You could tilt AB in any direction, and the perpendicular bisector would still meet it at a right angle and split it into two equal halves.
Worked construction 1: the perpendicular bisector of an 8 cm segment
- Draw AB: use a ruler to make an 8 cm segment.
- Choose a compass opening: select a radius greater than 4 cm, for example 5 cm.
- Draw arcs from A: with compass point at A, draw one arc above and one below AB.
- Keep the same opening at B: draw matching arcs from B to intersect the earlier arcs at points P and Q.
- Join P and Q: draw a straight line through them. It crosses AB at midpoint M and meets AB at 90°.
Why does this work? P was drawn 5 cm from both A and B, and so was Q. Points equally distant from A and B lie on the perpendicular bisector of AB. The line through P and Q therefore identifies that bisector.
The radius must be more than half the original segment length so that the arcs meet in two points rather than merely touching. This is an important step students sometimes miss when they use a compass opening that is too small.
What does an angle bisector do?
An angle bisector begins at the angle’s vertex and divides the angle into two equal parts. If a 60° angle is bisected, each new angle measures 30°. If a 90° angle is bisected, each new angle measures 45°.
This is different from bisecting a segment. An angle bisector splits an angle; it need not divide a line segment into equal lengths.
Worked construction 2: bisect a 60° angle
Suppose ∠ABC is 60°, with B as the vertex. Place the compass point at B and draw one arc that crosses BA at P and BC at Q. This places P and Q at the same distance from B.
Now set the compass to the same new opening for both sides. Draw an arc from P and another from Q so they meet at R inside the angle. Draw ray BR. It bisects ∠ABC, giving ∠ABR = 30° and ∠RBC = 30°.
The equality of compass openings is crucial. If you quietly change the compass width between drawing the matching arcs, the construction may look almost correct while failing the equal-angle property.
A useful property of points on an angle bisector
A point on the interior bisector of an angle is equally distant from the two sides when distance is measured along perpendiculars to those sides. That is one reason angle bisectors matter in geometric constructions: they are not just decorative lines.
Compare this with a perpendicular bisector, whose points are equally distant from the two endpoints of the original segment. The words ‘distance from sides’ and ‘distance from endpoints’ are different, and they tell you which bisector is being discussed.
Worked construction 3: form a 3–4–5 triangle
Suppose a triangle has side lengths 3 cm, 4 cm and 5 cm. Draw its longest side AB as 5 cm. Set the compass at A with a 3 cm opening and draw an arc. From B, draw an arc with a 4 cm opening that crosses the first at C. Join AC and BC.
The finished triangle has AC = 3 cm, BC = 4 cm and AB = 5 cm. Because 3² + 4² = 5², its angle at C is a right angle. Here a compass construction has produced a familiar shape that can also be checked by Pythagoras’ theorem.
A student should note that side lengths alone determine the triangle’s shape up to reflection and rotation. Choosing the intersection above or below AB makes a mirror-image construction, not a different collection of side lengths.
A geometry kit matters, but reasoning matters more
A practical set includes a ruler, compass, protractor and set square. Use the ruler for distances, the compass for equal distances, the protractor for angle measurements and the set square for checking perpendicular relationships.
An accurate-looking sketch is not necessarily a correct geometric construction. A tutor can ask students to show the equal-radius arcs and explain what each crossing point guarantees. That turns a drawing exercise into mathematical reasoning.
What SEAB expects at the G1 level
The official 2027 SEAB G1 Mathematics K110 syllabus lists properties of perpendicular bisectors of line segments and angle bisectors, and construction of simple geometric figures from given data using compasses, rulers, set squares and protractors where appropriate (G1, Geometry and Measurement). The SEC G1 examination directory identifies Paper 1 as the geometry-containing paper, and the K110 scheme notes that candidates should have geometrical instruments for it.
The 2027 Mathematics code is K110. SEAB lists 4046 for reference to 2026 and earlier. Parents of students sitting their examinations in 2026 should use that year’s actual subject code and school instructions. The drawings described here are original teaching exercises, not official specimen-paper tasks.
Five checks before a student puts the instruments away
- Correct object: is the task to bisect a line segment or an angle?
- Stable compass opening: did both matching arcs use exactly the same radius?
- Intersecting arcs: do the arcs meet where required?
- Correct line: does the perpendicular bisector cross the midpoint at 90°, or the angle bisector split the angle equally?
- Visible method: are the construction arcs still available to demonstrate the reasoning?
A five-minute check at home
Draw a 6 cm segment and ask the learner to construct its perpendicular bisector using arcs. The midpoint should be 3 cm from each endpoint, and the crossing angle should be 90°. Next draw an 80° angle and construct its bisector; the two resulting angles should each measure about 40° when checked with a protractor.
If the learner is very accurate with a ruler but the compass arcs fail to meet, the tutor can check the opening, not necessarily the geometry concept. If the arcs meet correctly but the wrong points are joined, the issue is identifying the intended construction.
The unchanged eduKateSG three-pupil Clementi Mathematics reference describes an individual-feedback teaching model for Secondary 1. It is not proof of a current Secondary 4 G1 construction class, timetable or available place; confirm those details with the provider directly.
Frequently asked questions
Is a midpoint the same as a perpendicular bisector? No. A midpoint is a point halfway along a segment; the perpendicular bisector is a whole line through that midpoint at 90°.
Can an angle bisector be drawn by measuring half the angle? A protractor can help check the result, but the compass method shows how the equality is constructed.
Are these topics part of G1 K110? Yes. The syllabus explicitly names both properties and construction methods.
Continue with connected geometry lessons
Read angles, symmetry and similarity, Pythagoras and right-triangle trigonometry and the Paper 1 revision map.