Bisecting lines and angles: Year 7 angles and constructions

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

From Unit 4, lesson 2 (angles on a line and at a point): what is the complement of 35°?
Answer
55°. Complementary angles add up to 90°, and 90 − 35 = 55.
From Unit 4, lesson 4 (angles in a triangle): two angles of a triangle are 50° and 70°. What is the third angle?
Answer
60°. 180 − (50 + 70) = 60.
From Unit 1, lesson 1 (place value and decimals): what is 9 ÷ 2 as a decimal?
Answer
4.5. Half of 9 is 4 and one half, which is 4.5.

Words used in this lesson

bisect: to cut exactly in half. For example: Bisecting 80° makes two angles of 40°.
perpendicular bisector: a line that cuts a segment in half at a right angle. For example: The line through M in the first figure.
compasses: a drawing tool that keeps a fixed distance from a point. For example: Used to draw an arc.
arc: a part of a circle. For example: The curved marks the compasses leave.
midpoint: the point exactly halfway along a segment. For example: M in the first figure.
equidistant: the same distance from two or more things. For example: P is equidistant from A and B.

The compass keeps a distance

A mason in Konya has to find the exact middle of an arch opening before the keystone goes in. With two pegs and a rope cut to length, the mason swings the rope from each peg. The compass does the same job on paper: whatever you set it to, every point it draws is the same distance from its centre. Two circles with equal radii about A and B cross at points that are the same distance from A as from B.

Predict first: AB is 8 cm. Will arcs of radius 3 cm meet? What about 5 cm? Try to say how big the radius has to be before you slide.

Compass radius (centimetres) 5 cm

bisector drawnresult

5 cmPA and PB

4 cmAM and MB

With radius 5 cm the arcs meet at P above AB and at Q below it. P is 5 cm from A and 5 cm from B, so the line through P and Q cuts AB at its midpoint M, 4 cm from each end.

The figure needs JavaScript. In words: AB is 8 cm long. Draw an arc of the same radius from A and from B, large enough to cross: more than half of 8, so more than 4 cm. The arcs meet at two points, one above and one below AB. Both points are the same distance from A as from B. The line through them is the perpendicular bisector: it cuts AB at its midpoint, 4 cm from each end, at a right angle. With a radius of 4 cm or less the arcs cannot meet in two points.

Show me the steps

AB = 8 cm. Radius 5 cm: the arcs meet at P above and Q below. PA = PB = 5 cm. The line PQ cuts AB at M, 4 cm from A and 4 cm from B, at a right angle.
Radius 6.5 cm: the arcs still meet, further up and down. PA = PB = 6.5 cm. M is still 4 cm from each end: a bigger radius moves P and Q, not M.
Radius 4.5 cm: the arcs only just cross, close to AB. The line is the same line.
Radius 4 cm: each arc reaches only the midpoint. The arcs touch at one point, so there is no second point and no line.
Radius 3 cm: the arcs fall short of each other and do not meet. The radius must be more than half of AB.

Bisect means cut exactly in half. To bisect a line segment AB: set the compasses to more than half of AB, draw an arc from A and an arc of the same radius from B so that they cross above and below, then join the two crossing points with a ruler. That line cuts AB at its midpoint and at a right angle, so it is the perpendicular bisector. Every point on it is the same distance from A as from B.

To bisect an angle: draw one arc about the vertex across both arms. Draw two arcs of the same radius from the points where the first arc cuts the arms, so that they cross. The ray from the vertex through the crossing point splits the angle into two equal angles.

Bisecting an angle

The same idea works one step along: the second arcs are drawn from U and V, which are the same distance from the vertex, so the crossing point is the same distance from U as from V. O is also the same distance from U and V, so O and W both lie on the perpendicular bisector of UV. Fold along OW: U lands on V, so the two halves of the angle match. That is the link with the first construction. Change the angle and the radius of the second arcs.

Angle (degrees) 70°

Radius of the second arcs (centimetres) 4 cm

35°each half of the angle

arcs meetsecond arcs

4 cmUW and VW

The angle is 70°. Arcs of 4 cm from U and V meet at W, and the line OW splits the angle into two halves of 35° each.

The figure needs JavaScript. In words: put the compass point on the vertex O and draw one arc across both arms, at U and V. Then put the compass point on U and draw an arc, and on V an arc of the same radius, so that they cross at W. The line from O through W splits the angle into two equal halves. For a 70 degree angle each half is 35 degrees. If the second radius is too small the arcs do not meet.

Show me the steps

A 70° angle. The first arc about O cuts the arms at U and V, both 5 cm from O. Arcs of 4 cm from U and V meet at W. OW bisects the angle: 35° and 35°.
Make the angle 120° and the second radius 4.5 cm. The chord UV is 8.7 cm, so half of it is 4.3 cm, and 4.5 cm is enough: the arcs meet at W and each half is 60°.
Keep 120° and shorten the second radius to 3.5 cm. The chord UV is 8.7 cm long, so half of it is 4.3 cm, and 3.5 cm is too short: the arcs do not meet.
Make the angle 30° and the radius 2.5 cm. The chord UV is 2.6 cm, half of it is 1.3 cm, and 2.5 cm is enough: the arcs meet and each half is 15°.

A common mistake: “I used a different setting on the compasses for each arc, as long as they cross.” The arcs have to use the same radius, or the crossing point is not the same distance from both ends, and the line it makes will not bisect anything. Keep the compasses locked, and do not let the setting slip between arcs.

Check your own drawing. A construction on paper is drawn accurately, so here you do measure, to check it. Bisect a 7 cm segment: AM should be within 1 mm of 3.5 cm and the angle at M within 1° of 90°.

Worked example: bisect a line, then an angle

A segment AB is 9 cm long. Bisect it, then say how long each half is. Then bisect an angle of 80° and say what each half measures.

Set the compasses to more than half of AB, which is 4.5 cm. A radius of 6 cm works.
Draw arcs of 6 cm from A and from B, above and below. Join the two crossing points.
The line cuts AB at its midpoint M, so AM = MB = 9 ÷ 2 = 4.5 cm, at a right angle.
For the 80° angle, the bisector makes two equal angles: 80 ÷ 2 = 40°. 4.5 cm and 40°

Your go. The first two steps are given. Bisect an angle of 100°, then bisect one of the two halves again. Find the size of the smallest angle you make.

The first bisector makes two angles of 100 ÷ 2 = 50°.
Take one 50° angle. Draw an arc about the vertex, then two equal arcs from where it cuts the arms of that angle.
Now work out the size of each part of the 50° angle yourself, then check.

Check the full answer

The second bisector splits 50° into two angles of 50 ÷ 2 = 25°. The smallest angle is 25°, and the other half of the original angle is still 50°.

Check before you go on

1. You set your compasses to 3 cm to bisect a line segment AB that is 8 cm long, and draw arcs from A and from B. Why does the construction fail?

The arcs cannot reach each other, because 3 + 3 = 6 is less than 8. The arcs meet, but the line is not perpendicular. A radius of 3 cm always works, because bisecting only needs equal radii. It fails because the compasses must be set to exactly 4 cm.

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A segment AB is 11 cm long. Its perpendicular bisector cuts AB at M. How long is AM, in centimetres, to one decimal place?

3. A point P lies on the perpendicular bisector of AB. PA = 7.2 cm. What is PB?

3.6 cm 7.2 cm 14.4 cm It cannot be found.

Level 2 · Building

4. An angle of 78° is bisected. What is each half, in degrees?

5. A bisector divides an angle into two angles of 26° each. How big was the original angle, in degrees?

6. Which description gives the steps for bisecting an angle in the right order?

Arc about the vertex across both arms; two equal arcs from those two points so they cross; join the vertex to the crossing. Join the two points where an arc cuts the arms; measure the line; mark the middle; join it to the vertex. Draw two arcs about the vertex with different radii; join the points where they cross. Measure the angle with a protractor and draw a line at half the size.

Level 3 · Stretch

7. An angle of 120° is bisected. One of the halves is then bisected again. What is the size of the smallest angle made, in degrees?

8. AB is 12 cm. M is the midpoint of AB, and N is the midpoint of AM. How long is NB, in centimetres?

Something to think about

Bisect a 12 cm segment, then bisect each half, and keep going. How long is each piece after 1, 2, 3 and 4 rounds of bisecting? After how many rounds is a piece shorter than 1 cm? Which pattern from Unit 1 does the number of pieces follow?

One way to think about it

The pieces are 6 cm, 3 cm, 1.5 cm and 0.75 cm: the length is halved each round, 12 ÷ 2, 12 ÷ 4, 12 ÷ 8, 12 ÷ 16. After 4 rounds a piece is under 1 cm. The number of pieces is 2, 4, 8, 16, the powers of 2 from lesson 2 of Unit 1.

Exit ticket

9. Zaid draws an arc of radius 5 cm from A and an arc of radius 6 cm from B, so that they cross at X and Y. He joins X and Y. Is XY the perpendicular bisector of AB?

Yes, because any two arcs that cross give the bisector. No: points on the bisector are the same distance from A and B, which needs equal radii. Yes, but only if AB is less than 11 cm. No, because arcs must be drawn from the middle of AB.