Question

Construct a parallelogram ABCD in which AB = 5.6 cm, AC = 6 cm and BD = 6.2 cm.

Answer 1

Given:

AB = 5.6 cm

AC = 6.0 cm

BD = 6.2 cm

Step-wise calculation:

1. In a parallelogram, the diagonals bisect each other.

So, let O be their intersection.

Then `AO = (AC)/2`

= `(6.0)/2`

= 3.0 cm

And `BO = (BD)/2`

= `(6.2)/2`

= 3.1 cm

2. Check that triangle AOB is constructible:

AO + BO

= 3.0 + 3.1

= 6.1 > AB

= 5.6

So, the three lengths 3.0, 3.1, 5.6 form a triangle (triangle inequality holds).

Construction steps (compass and straightedge):

1. Draw segment AB = 5.6 cm. Label the left endpoint A and the right endpoint B.

2. With centre A and radius 3.0 cm (AO), draw an arc.

3. With centre B and radius 3.1 cm (BO), draw an arc that intersects the arc from step 2.

Mark one intersection as O (choose the intersection on the same side of AB where you want the parallelogram to lie).

4. Join A to O and B to O (optional, to see triangle AOB).

5. Construct C as the point on the line AO such that O is the midpoint of AC: extend line AO beyond O and with centre O and radius AO = 3.0 cm mark point C on that extension (so OC = 3.0 cm).

6. Construct D as the point on the line BO such that O is the midpoint of BD: extend line BO beyond O and with centre O and radius BO = 3.1 cm mark point D on that extension (so OD = 3.1 cm).

7. Join A–B, B–C, C–D and D–A to complete parallelogram ABCD.

Verification:

By construction

AO = OC = 3.0 cm 

⇒ AC = AO + OC = 6.0 cm

By construction

BO = OD = 3.1 cm

⇒ BD = BO + OD = 6.2 cm.

AB was drawn 5.6 cm.

Because O is the midpoint of both diagonals, ABCD is a parallelogram with the required measurements. 

This is the standard method for “one side and both diagonals given.”

Parallelogram ABCD is constructed with AB = 5.6 cm, AC = 6 cm and BD = 6.2 cm.