IGCSE Maths / Geometric constructions
Constructing triangles with compasses
Edexcel International GCSE Maths (4MA1), chapter 13: Geometric constructions, specification point 4.5b. Free lesson.
Read the revision notes · IGCSE Maths past papers
Construct a triangle from three side lengths by using a ruler and compasses to locate its third corner. The arcs show that the point was constructed, so they must remain visible for the marks.
What it means to construct a triangle
To construct a point is to fix its position with a ruler and compasses, rather than by measuring. The curved pencil marks made by the compasses are called arcs. They show how the point was found, so both arcs must still be visible when you finish. A triangle with the right side lengths but no construction arcs has been measured, not constructed. Do not rub out arcs that extend beyond the triangle.
How to construct a triangle from three sides
Start with the longest side as the base. The other two sides set the distances from each end of the base, so their arcs cross at the only point that has both required distances.
Check the 6 cm, 8 cm, 10 cm triangle
The construction fixes a triangle with sides 6 cm, 8 cm and 10 cm. You can check its angle at A by squaring the two shorter sides and comparing their total with the longest side squared. Since the totals match, the angle opposite the longest side is a right angle. 6² + 8² = 36 + 64 = 100 10² = 100 The longest side is BC, so the right angle is opposite it, at A. Therefore, angle BAC = 90°.
When three lengths cannot make a triangle
Not every set of three lengths can form a triangle. The two shorter sides must add up to more than the longest side. Otherwise, the arcs from the ends of the base cannot reach each other. For 3 cm, 4 cm and 12 cm, the shorter sides total 3 + 4 = 7 cm, which is less than 12 cm. The arcs therefore do not cross, so no triangle can be constructed.
Avoid measuring the third corner
Measuring a position for the third corner might give a triangle with roughly the right side lengths, but it does not show that the point was constructed. The crossing arcs are the evidence the mark scheme looks for. Set each compass width from the ruler, then reset it for the second arc. The third corner is the point where the two arcs cross. Keep both arcs visible, even if they overshoot the sides.
Use the same check for the largest angle
For a triangle with sides 5 cm, 12 cm and 13 cm, compare the sum of the squares of the shorter sides with the square of the longest side. The matching totals show that the largest angle, opposite the 13 cm side, is 90°. 5² + 12² = 25 + 144 = 169 13² = 169