Question

Using ruler and compass only, construct a rhombus ABCD in which AB = 4 cm and ∠BCD = 120°.

Answer 1

Given:

AB = 4 cm, all sides of a rhombus are equal

∠BCD = 120°

Step-wise calculation (ruler and compass construction):

1. Draw a straight line and mark point C. On that line, with the ruler, mark point B so that BC = 4 cm.

We may start with BC = 4 cm because all sides of the rhombus equal AB; starting from BC is convenient since the given angle is at C.

2. At C construct a 60° angle by the standard equilateral-triangle arc method:

With center C and an arbitrary radius r, draw an arc that cuts ray CB at E.

With the same radius r and center E draw an arc that intersects the first arc at F.

Join C to F. Ray CF makes 60° with CB.

3. To get 120° at C: draw the straight line through C and F; take the ray from C on the side opposite to ray CF, this ray is CD and makes 180° – 60° = 120° with CB.

Thus, ∠BCD = 120°.

4. On the ray CD, measure off CD = 4 cm. 

Use the compass set to 4 cm and mark point D.

5. With the compass centered at B and radius 4 cm, draw an arc.

With the compass centered at D and the same radius 4 cm, draw another arc.

Let the appropriate intersection of these two arcs be point A. 

Choose the intersection that gives a convex quadrilateral ABCD.

Because AB and AD are both constructed as radii 4 cm from B and D, we get AB = AD = 4 cm.

6. Join A to B and A to D with the ruler. Now the quadrilateral ABCD has

AB = BC = CD = DA = 4 cm, all sides equal.

∠BCD = 120° by construction.

Therefore, ABCD is a rhombus with the required data.

The constructed quadrilateral ABCD with vertices in order ABCD is a rhombus with side length 4 cm and ∠BCD = 120°.