A triangle has six measurements: three sides and three angles. You do not need all six to draw it. You need exactly three, and they have to be the right three. Give somebody fewer than three and they can draw infinitely many different triangles. Give them the wrong three and they still cannot pin it down. So before you touch the compass, ask: which three have I been given?
The three cases you will meet
| Given | Short name | How you build it |
|---|---|---|
| Three sides | SSS | Draw one side. Swing an arc of the second length from one end and an arc of the third from the other. They cross at the top corner. |
| Two sides and the angle between them | SAS | Draw one side. Construct the given angle at one end. Measure the second side along that new arm. Join up. |
| Two angles and the side between them | ASA | Draw the side. Construct one angle at each end. The two new arms cross at the third corner. |
Every one of them starts the same way: draw one side, full size, with a ruler. That side is the foundation and it is the only measurement you are allowed to take with a ruler. After that the compass does the work. What changes between the three cases is only what you do at the ends of that first side, and in every case the third corner appears where two things cross.
Suppose you are told all three angles: 60, 60 and 60. How big is the triangle? You cannot say. A tiny equilateral triangle and an enormous one both have those angles. Three angles fix the shape but say nothing about the size, so there are infinitely many triangles that fit. This case is called AAA and it is not a construction case at all. At least one of your three pieces of information has to be a length.
When the arcs refuse to meet
Try to build a triangle with sides 3 cm, 4 cm and 9 cm. Draw the 9 cm side, then swing a 3 cm arc from one end and a 4 cm arc from the other. They never touch. This is not a mistake in your drawing. That triangle does not exist, because 3 plus 4 is only 7, and 7 is less than 9. The two short sides cannot reach across the long one however you angle them.
Any two sides of a triangle must add up to more than the third side. Check the largest side against the sum of the other two and you will know before you start whether the triangle is possible. 3, 4, 9 fails. 3, 4, 6 works, because 3 plus 4 is 7 and 7 is more than 6. This is worth thirty seconds of checking, because it saves you from a construction that was never going to close.
In the last unit you used SSS backwards: you forced three sides to match so that two angles would come out equal. Here you use it forwards: three lengths are given and you build the one triangle those lengths allow. It is the same fact both times. Three sides fix a triangle completely, so if you can lay down three sides you have laid down the whole thing, angles included.
SAS says two sides and the angle BETWEEN them. The word between is doing real work. If you are given two sides and an angle that is not between them, the information sometimes fits two completely different triangles, so it does not pin one down. In Grade 7 you will always be given the angle between the two sides, but it is worth knowing why the question is always worded that way.
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Construct triangle XYZ with XY = 6 cm, YZ = 5 cm and ZX = 4 cm
| Step | What you do |
|---|---|
| 1 | With a ruler, draw XY exactly 6 cm long. |
| 2 | Set the compass to 4 cm against the ruler. With X as centre, draw an arc above XY. |
| 3 | Set the compass to 5 cm. With Y as centre, draw an arc cutting the first one at Z. |
| 4 | Join XZ and YZ. Triangle XYZ is complete. |
Construct triangle PQR with PQ = 6 cm, angle P = 60° and PR = 5 cm
| Step | What you do |
|---|---|
| 1 | Draw PQ = 6 cm with a ruler. |
| 2 | At P, construct 60° with the compass: arc from P cutting PQ, then step the same radius round once. |
| 3 | Draw the arm from P through that mark, longer than 5 cm. |
| 4 | Set the compass to 5 cm. From P, cut the arm at R. |
| 5 | Join RQ. Triangle PQR is complete. |
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A triangle has six measurements but you need exactly three of the right kind to construct it.
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