Question

Construct a rhombus ABCD, when BD = 6 cm, ∠ABC = 120°.

Answer 1

Given:

• Diagonal BD = 6 cm
• ∠ABC = 120°

We need to construct a rhombus ABCD with these conditions.

Steps of construction:

1. Draw line segment BD = 6 cm.
2. Find the midpoint O of BD by drawing the perpendicular bisector of BD.
3. Since ABCD is a rhombus, the diagonals bisect each other at right angles. So, draw a perpendicular bisector to BD at O.
4. Let the length of diagonal AC be unknown. Denote AO = OC = x cm half of AC.
5. At point B, we know ∠ABC = 120°. This angle is between sides AB and BC.
6. Since ABCD is a rhombus, all sides are equal, so AB = BC = AD = CD.
7. By properties of rhombus, the angle between adjacent sides corresponds to ∠ABC = 120°. Using this and the known diagonal BD, we can find the length of side AB and the half diagonal AO.
8. Using the triangle ABO:
   • BD is given BD = 6 cm
   • O is midpoint of BD so BO = 3 cm
   • At B, angle between side AB and BO can be related to the known angle 120°
9. Using trigonometry or construction:
   • Construct BD = 6 cm.
   • At B, construct an angle of 120° angle ABC.
   • From B, set AB = BC = length equal to be determined so that the figure closes and forms rhombus.
10. Another approach is:
    • Draw BD = 6 cm.
    • At B, construct angle ABC = 120°.
    • With B as center and radius equal to the length of side AB to be found, draw arc to intersect the perpendicular bisector of BD at points A and C.
    • Since diagonals bisect each other at right angles, construct perpendicular bisector of BD; intersections with arcs give points A and C.
    • Join points A and C.
    • Join sides AB, BC, CD and DA.

Note: The length of AB can be calculated using geometric relations:

Using the law of cosines in triangle ABC:

AB = BC = s

∠ABC = 120°

Apart from this, using diagonal relations and given BD = 6 cm, AB can be found via formulas but since this is a construction, usually measurement or compass used.

Construct BD = 6 cm.

Construct the perpendicular bisector of BD and mark midpoint O.

At B, construct angle ABC = 120°.

Using compass and intersection of arcs from B and symmetry about O, locate points A and C.

Join A to B, B to C, C to D and D to A to complete rhombus ABCD with BD = 6 cm and ∠ABC = 120°.

Thus, ABCD is the required rhombus satisfying the given conditions.