Mathematics • Year 7 • Unit 3 • Lesson 19
Constructions: Bisecting Angles and Lines
Build fluency with the three classical constructions: perpendicular bisector of a segment, angle bisector, and perpendicular from a point to a line. Two tools only: a straight-edge and a pair of compasses. Do the constructions on paper, describe them clearly in writing on this worksheet.
1. I do, fully worked example
Read every line. Each step shows why the construction works.
Problem. Describe the four steps for constructing the perpendicular bisector of a segment AB, and explain why the construction works.
Step 1, Set the compass radius.
Open the compass to MORE than half the length of AB. (About 3/4 of AB is safe.)
Reason: if the radius is less than half AB, the arcs from A and B won't reach each other and won't intersect.
Step 2, Arcs from A.
Place compass point on A. Draw an arc above AB and a second arc below AB.
Reason: every point on these arcs is the same distance (the radius) from A.
Step 3, Arcs from B (same radius, don't change the compass!).
Move compass point to B. Draw arcs above and below that CROSS the first two arcs.
Reason: every point on these arcs is the same distance from B. The two crossing points are therefore equidistant from BOTH A and B.
Step 4, Join the two crossings.
Use the straight-edge to draw a line through both intersection points.
Reason: every point on this line is equidistant from A and B. That's the definition of a perpendicular bisector, and the line crosses AB at right angles through the midpoint.
Answer: Set radius wider than half AB → arc from A above and below → arc from B above and below (same radius) → join the two crossings.
2. We do, fill in the missing steps
Same structure as Section 1, with the working faded. Fill in each blank. 4 marks
Problem. Describe the five steps to construct the bisector of an angle ∠ABC (vertex at B).
Step 1, Centre on the vertex. Place the compass point at _______ . Choose any radius.
Step 2, Arc across both arms. Draw an arc that crosses BOTH sides of the angle. Label the crossing on BA as _______ and the crossing on BC as _______ .
Step 3, Arc from P. WITHOUT changing the radius, place the compass point on _______ and draw an arc INSIDE the angle.
Step 4, Arc from Q. Same radius from _______ . Draw an arc that crosses the previous one. Label the crossing _______ .
Step 5, Draw the bisector. Use the straight-edge to draw a ray from _______ through _______ . This ray is the angle bisector, it cuts ∠ABC into two equal halves: ∠ABR = ∠RBC.
3. You do, independent practice
Answer each question. Some are written description; some are calculation. The first four are foundation, the middle two are standard, and the last two are extension.
Foundation, recall
3.1 Name the TWO tools used in a classical construction. 1 mark
3.2 A perpendicular bisector cuts a segment into two parts of length ____ and crosses it at ___° . 1 mark
3.3 Bisecting an angle of 80° creates two angles, each of ___° . 1 mark
3.4 What does it mean for a line to be "perpendicular" to another? 1 mark
Standard, apply the constructions
3.5 Bisecting a right angle gives an angle of ___° . Explain why. 2 marks
3.6 If the perpendicular bisector of AB is constructed and AB = 10 cm, what is the distance from the midpoint of AB to each endpoint? 2 marks
Extension, push your thinking
3.7 Explain in your own words why the angle bisector construction works. Use the fact that triangles △BPR and △BQR are congruent by SSS. 2 marks
3.8 Bisecting an angle TWICE (i.e. bisect, then bisect one of the halves) divides the original angle into how many equal parts? If the original angle is 120°, what is the smallest angle produced? 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (angle bisector)
Step 1: compass at B.
Step 2: crossings labelled P (on BA) and Q (on BC).
Step 3: compass on P.
Step 4: compass on Q; crossing labelled R.
Step 5: draw ray from B through R.
3.1, Two tools
Straight-edge (ruler used only for its edge) and a pair of compasses. NO protractor.
3.2, Perpendicular bisector facts
Cuts into two equal halves; crosses the segment at 90° (right angle).
3.3, Bisecting 80°
Each new angle = 80 ÷ 2 = 40°.
3.4, Perpendicular
"Perpendicular" means at a right angle (90°) to another line.
3.5, Bisecting a right angle
Right angle = 90°. Bisected: each new angle = 90 ÷ 2 = 45°. Because the bisector cuts the angle exactly in half, the two new angles are equal, both 45°.
3.6, Midpoint distance
The perpendicular bisector passes through the midpoint. AB = 10 cm, so each half = 5 cm from the midpoint to each endpoint.
3.7, Why the angle bisector works
The compass guarantees that BP = BQ (same arc from B) and PR = QR (same arc from each of P and Q). Also BR = BR (same side, shared by both triangles). So in triangles △BPR and △BQR, all three sides are equal, the triangles are congruent by SSS. Corresponding angles in congruent triangles are equal, so ∠PBR = ∠QBR. That means BR cuts the original angle into two equal halves, it's the angle bisector.
3.8, Bisecting twice
First bisection: original angle ÷ 2 → 2 equal pieces. Bisect ONE of those pieces → 2 more pieces. Total pieces: 1 unbisected half + 2 quarter pieces = 3 pieces in total (not equal! one is half, two are quarters).
If the original is 120°: first bisection gives two 60° halves. Bisecting one half gives two 30° quarters. So the smallest angle produced = 30°.
If both halves were bisected, the result would be 4 equal quarters of 30° each, but the question only asks about ONE further bisection.