Question

Construct a rhombus ABCD in which AB = 4 cm and AC = 6.2 cm.

Answer 1

Given:

AB = 4.0 cm

AC = 6.2 cm

Step-wise calculation:

1. In a rhombus, the diagonals bisect each other at right angles.

So, if O is the intersection of diagonals then

AO = OC

= `(AC)/2`

= `6.2/2`

= 3.10 cm

2. Let BO = DO = x   ...(Half of BD) 

In right triangle AOB:

AO² + BO² = AB² 

(3.10)² + x² = 4.0²

9.61 + x² = 16x²

= 16 – 9.61

= 6.39

`x = sqrt(6.39)` 

= 2.528 cm

3. Therefore, BD = 2x

= 2 × 2.528

= 5.056 cm (≈ 5.06 cm) 

This use of the diagonals and the construction-by-arcs method is the standard approach when one side and one diagonal are given.

Construction steps (compass and straightedge):

1. Draw segment AC = 6.2 cm and mark endpoints A and C.

2. Locate the midpoint O of AC, measure 3.10 cm from A or construct the perpendicular bisector.

3. Through O, draw a line perpendicular to AC.

4. With center O and radius = BO = 2.528 cm (≈ 2.53 cm), mark two points on the perpendicular, one on each side of AC; label them B and D.

Equivalently: Draw arcs of radius AB = 4 cm centered at A and C; their intersections are B and D. This is the common construction given for “one side and a diagonal given”.

5. Join ABCD. The resulting quadrilateral is the required rhombus.