Three lengths, and where to start
The task is to construct a triangle whose sides are 8 cm, 6 cm and 5 cm long, with a ruler to set lengths and a compass. Only one side can be drawn straight away, because it needs no angle: rule a base AB 8 cm long. The longest side is the usual choice, and any of the three would do.
The third corner, C, is not known yet. What is known about it is two distances: C must be 6 cm from A and 5 cm from B.
The base AB, ruled 8 cm long.
An arc for each remaining side
Set the compass to 6 cm against the ruler, put its point on A, and draw an arc above the base. Every point on that arc is 6 cm from A, so C is somewhere on it. The arc alone does not say where.
Now reset the compass to 5 cm, put its point on B, and draw an arc that crosses the first one. Every point on this arc is 5 cm from B. The crossing is on both arcs, so it is 6 cm from A and 5 cm from B at once. It is the only point above AB that is both, so it is C.
An arc of 6 cm from A. Corner C is somewhere on it.
An arc of 5 cm from B crosses it at C, the one point 6 cm from A and 5 cm from B.
Finish with the ruler
Rule AC and BC. Measure them to check: AC should be 6 cm and BC 5 cm. Leave the arcs on the page; they show how C was found.
The finished triangle, with sides of 8 cm, 6 cm and 5 cm.
Only one triangle
The two arcs cross below AB as well. The triangle built on that crossing is the mirror image of the first one, and turned over it fits exactly on top of it. Two shapes that fit exactly on top of each other, after any turning or flipping, are called congruent: they have the same shape and the same size.
So three side lengths leave no freedom at all. Everyone who builds a triangle from 8 cm, 6 cm and 5 cm builds a congruent triangle, with the same three angles too. Three angles do not do the same: a triangle with angles of 60°, 60° and 60° can be any size.
The width goes with the side
In the bisector constructions the compass kept one width for both arcs. Here it must not. Keeping 6 cm for the arc from B would build a side of 6 cm from B, not 5 cm. Each arc is set to the length of its own side: 6 cm from A, because AC is 6 cm, and 5 cm from B, because BC is 5 cm.
Ruling a 6 cm side from A at a guessed angle does not work either. The 5 cm side then has to be forced to fit, and it will not reach B exactly.
When the arcs do not meet
Try sides of 8 cm, 3 cm and 4 cm. Rule the 8 cm base and draw an arc of 3 cm from A and an arc of 4 cm from B. Along the base the two arcs reach only 3 + 4 = 7 cm, which is 1 cm short of 8 cm, and they never meet. There is no triangle.
With 8 cm, 3 cm and 5 cm, 3 + 5 = 8 exactly, and the two arcs meet at a single point on the base itself. The "triangle" is flat: its three corners lie on one straight line.
The reason is that the straight segment AB is the shortest path from A to B. A path from A to a corner C and on to B is longer unless C lies on the segment AB. So in every triangle, any two sides add up to more than the third side. This is called the triangle inequality. For 8, 6 and 5 it holds: 6 + 5 = 11, which is more than 8.
A and B are 8 apart. The circle of radius 3 around A and the circle of radius 4 around B stop 1 short of each other, so no point is 3 from A and 4 from B.
Worked example: Possible Lengths of the Third Side
Question Two sides of a triangle are 7 cm and 12 cm long. The third side is a whole number of centimeters. What is the greatest possible length of the third side, and how many different whole-number lengths are possible?
1.Any two sides of a triangle add to more than the third, or the two short ones could not meet.
Swing the 7 cm side about C and watch the third side change. 2.Third side < 7 + 12 = 19 cm; and 7 + third > 12, so third side > 12 − 7 = 5 cm.
Fully open it is 7 + 12 = 19 cm; fully closed, 12 − 7 = 5 cm. 3.So the third side is longer than 5 cm and shorter than 19 cm; 5 and 19 themselves give a flat, closed-up figure, not a triangle.
Neither end is a triangle: the three sides lie flat along one line. 4.The greatest whole number below 19 is 18: greatest possible length 18 cm.
Greatest whole number below 19: 18 cm. 5.The whole numbers from 6 to 18: 18 − 6 + 1 = 13 possible lengths.
Whole numbers from 6 to 18: 13 of them.
Answer: 18 cm; 13 possible lengths
Common mistakes
- Including 19 and 5: at those lengths the three sides lie flat along one line.
- Counting 18 − 6 = 12 lengths: a run from 6 to 18 has 13 numbers, both ends included.