Constructing triangles

Why a compass helps

Suppose we want a triangle with every side 4 cm, an equilateral triangle. With only a marked ruler, we draw a base AB of 4 cm and then try to mark a point C that is 4 cm from A, checking whether it is also 4 cm from B, again and again. A compass removes the guesswork. All the points 4 cm from A lie on a circle of radius 4 cm around A, and all the points 4 cm from B lie on a circle around B. The point we want lies on both, so it is where the two arcs cross.

Three sides given

The same idea works for any three sides, say 4 cm, 5 cm and 6 cm.

1. Draw the base AB = 4 cm.
2. From A, draw an arc of radius 5 cm, long enough to cross the next one.
3. From B, draw an arc of radius 6 cm that cuts the first arc.
4. The crossing point is C. Join AC and BC.

Because C lies on both arcs, AC is 5 cm and BC is 6 cm. Full circles are not needed, only arcs that are long enough to meet. Triangles such as 4, 4, 6 or 3.5, 3.5, 3.5 are built the same way; with two equal sides the triangle is isosceles, and with three it is equilateral.

Not every three lengths work: if the longest is not less than the sum of the other two, the arcs never meet. That rule is explained in triangle inequality and angle sum property.

Two sides and the included angle

The included angle is the angle between the two given sides. To construct ∆ABC with AB = 5 cm, AC = 4 cm and ∠A = 45°:

1. Draw AB = 5 cm.
2. At A, draw an arm making 45° with AB.
3. Mark C on that arm so that AC = 4 cm.
4. Join BC.

A triangle can be made this way for any included angle less than 180°; only a straight angle or more makes it impossible.

Two angles and the included side

The included side is the side shared by the two given angles. To construct ∆ABC with AB = 5 cm, ∠A = 45° and ∠B = 80°, draw AB, draw 45° at A and 80° at B, and the two new arms meet at C. Here the two angles must add up to less than 180°, otherwise the arms do not meet.

Constructing an altitude

An altitude from A is the perpendicular from A to the opposite side BC. Drawing it with a ruler alone is not accurate, so we use a set square: lay the ruler along BC, rest one edge of the set square’s right angle on the ruler, slide it until the upright edge touches A, and draw along that edge. If the triangle is obtuse at B or C, extend BC first, since the altitude then falls outside the triangle.

Why it matters

Careful constructions turn the facts of geometry into accurate drawings, and they help us test ideas, such as whether a triangle exists. The kinds of triangle these constructions produce are in types of triangles. Answers to every question are in the NCERT solutions for Chapter 7; see also the full Chapter 7 notes.