I know the three sides of a triangle. Can I draw it exactly with a ruler and compasses?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Draw a triangle from three sides using compasses
Draw a triangle from two sides and the angle between them
Cut an angle exactly in half using compasses
A construction is a drawing made with a ruler and compasses. Compasses draw circles. A piece of a circle is an arc. You leave every arc on the page, so anyone can see how you drew it.
The pictures in this chapter are smaller than real size. Every length in one picture uses the same scale. Draw your own at full size.
Draw it first
A triangle from three sides
1 Draw the base BC, 5 cm long, with a ruler.
2 Set the compasses to 3 cm. Put the point on B and draw an arc.
3 Set the compasses to 4 cm. Put the point on C and draw an arc that crosses the first. Mark the crossing A.
4 Join A to B and A to C with the ruler. Leave the arcs in.
Worked example
Draw a triangle with sides of 6 cm, 5 cm and 4 cm.
Draw the longest side, 6 cm, as the base BC.
Set the compasses to 4 cm. Put the point on B and draw an arc.
Set the compasses to 5 cm. Put the point on C and draw an arc that crosses the first.
Mark the crossing A. Join A to B and A to C.
Check with the ruler. AB is 4 cm and AC is 5 cm.
Answer: A triangle with sides of 6 cm, 5 cm and 4 cm
The arcs stay in. They show the compasses were set to 4 cm and 5 cm.
Do the arcs always meet? No. The two shorter sides together must be longer than the longest side. If they are not, the arcs miss each other.
Sides of 7 cm, 3 cm and 2 cm make no triangle. 3 + 2 is less than 7.
Draw it first
A triangle from two sides and the angle between them
1 Draw the base BC, 6 cm long.
2 Put the protractor on B. Mark 50° from the base and draw a line through the mark.
3 Set the compasses to 4 cm. Put the point on B and draw an arc across the line. Mark the crossing A.
4 Join A to C.
Worked example
Draw a triangle. Two sides are 6 cm and 4 cm. The angle between them is 50°.
Draw the base BC, 6 cm long.
At B, use a protractor to mark 50°. Draw a line through the mark.
Set the compasses to 4 cm. Put the point on B and draw an arc across the line.
Call the crossing A. Join A to C.
The angle at B is the angle between the 6 cm side and the 4 cm side.
Answer: A triangle with sides of 6 cm and 4 cm and an angle of 50°
The angle at B is the angle between the two sides you were given.
Compasses can also cut an angle in half. We say they bisect the angle. The line that does this is the angle bisector.
Draw it first
Cut an angle of 80° in half
1 Draw an angle of 80°. Its corner is O.
2 Put the compass point on O. Draw an arc that crosses both arms, at P and at Q.
3 Keep the compasses the same. Draw an arc from P and an arc from Q. They cross at R.
4 Draw a line from O through R. It cuts the angle into two halves of 40°.
See it three ways
With things
Take two strings, one 3 cm and one 4 cm. Hold their ends on the two ends of a 5 cm line. Pull the loose ends together. They meet at one point.
As a picture
In symbols
3 + 4 is more than 5, so the strings meet. 2 + 3 is less than 7. Strings of 2 cm and 3 cm cannot meet over a 7 cm line.
Try it
Draw a triangle with compasses
You need: a sheet of paper, a ruler, a pair of compasses, a pencil.
Be safe: Compasses have a sharp point. Hold them by the top. Keep the point away from your face and from other people. Close them and cover the point when you put them down.
Draw a line 7 cm long. Call its ends B and C.
Set the compasses to 5 cm. Put the point on B and draw an arc above the line.
Set the compasses to 6 cm. Put the point on C and draw an arc that crosses the first.
Mark the crossing A. Join A to B and A to C. Measure AB and AC with the ruler.
What you will see: AB is 5 cm and AC is 6 cm. Anyone who draws sides of 7 cm, 6 cm and 5 cm gets the same shape.
If you cannot try it, this is what happens.
A wrong idea many people have
You can draw a triangle with any three lengths.
Why it feels right: Any three sticks seem as if they should make a triangle.
The test that breaks it: Try sides of 2 cm, 3 cm and 7 cm. Draw the 7 cm base and the two arcs. They never meet.
What is true: The two shorter sides together must be longer than the longest side.
Key words
construction
a drawing made with a ruler and compasses, with the arcs left in.
compasses
a tool that draws circles and arcs. It has a sharp point.
arc
a piece of a circle.
bisect
to cut exactly in half. The line that bisects an angle is the angle bisector.
Back to your guess
The answer is A: Yes. Two arcs show where the third corner goes.
Draw one side as the base. Set the compasses to the second side and draw an arc from one end. Set them to the third side and draw an arc from the other end. The arcs cross at the third corner.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
You draw a triangle with sides of 5 cm, 3 cm and 4 cm. The base BC is 5 cm. You draw an arc from B with the compasses set to 3 cm. What do you set them to for the arc from C?
See the answer
4 cm
The base uses 5 cm and the arc from B uses 3 cm. The third side is 4 cm, so set the compasses to 4 cm.
If you chose A: That is the length of the base. The arc from C uses the third side. Go to M3.5.5.
If you chose B: That is the setting you already used for the arc from B. The arc from C uses the side that is left. Go to M3.5.5.
Can you draw a triangle with sides of 7 cm, 3 cm and 2 cm?
See the answer
No. The arcs do not meet.
3 cm and 2 cm together are 5 cm, which is less than 7 cm. The arcs cannot cross.
If you chose A: Not any three. The two short sides must be longer together than the long side. Here 3 + 2 is less than 7. Go to M3.5.5.
If you chose C: The ruler is not the problem. The sides are too short to reach each other, whatever tool you use. Go to M3.5.5.
Which three lengths can be the sides of a triangle?
See the answer
5 cm, 6 cm, 10 cm
5 + 6 = 11, which is more than 10. The other two sets have two sides that add to 7 or 4, not more than the third.
If you chose B: 2 + 2 is exactly 4. The arcs only touch on the base, so you get a flat line, not a triangle. Go to M3.5.5.
If you chose C: 3 + 4 is only 7, which is less than 8. The arcs would not meet. Go to M3.5.5.
A triangle has a base BC of 6 cm and a side BA of 4 cm. The angle at B is 50°. How do you find the corner A?
See the answer
Draw a line at 50° from B. Set the compasses to 4 cm and draw an arc about B across the line.
The corner A is on the 50° line, and 4 cm from B. The arc about B at 4 cm crosses the line at A.
If you chose A: The 6 cm is the base BC, which is already drawn. The arc uses the side BA. Go to M3.5.5.
If you chose B: That draws a triangle from three sides. Here you know an angle, so you draw a line at 50° from B first. Go to M3.5.5.
The arcs are drawn from P and from Q and they cross at R. The line from O through R cuts the angle in half. What is each half?
See the answer
50°
The line through R cuts the angle exactly in half. 100° ÷ 2 = 50°.
If you chose A: That is double the angle. Cutting an angle in half makes the parts smaller, not bigger. Go to M3.5.5.
If you chose C: That is a quarter of 100°. Cutting in half means dividing by 2. Go to M3.5.5.
An angle of 84° is cut in half with compasses. How big is each half?
See the answer
42°
Each half is 84° ÷ 2 = 42°.
From an earlier chapter
A right angled triangle has an angle of 28°. What is the third angle?
See the answer
62°
The three angles add to 180°. 90° + 28° = 118°. Then 180° − 118° = 62°.
Keep it
Copy this into your exercise book.
Three sides: draw the base, then two arcs that cross at the third corner. Two sides and an angle: mark the angle, then use an arc for the length. Arcs from P and Q cut an angle in half.
Draw: A base BC with two arcs crossing at A. Join A to B and A to C. Write the three side lengths in cm.
