Question

Construct a parallelogram ABCD in which AB = 4.5 cm, AD = 5 cm and ∠A = 75°.

Answer 1

Given:

Parallelogram ABCD with AB = 4.5 cm, AD = 5.0 cm and ∠A = 75°.

Construction uses the standard method: Draw two adjacent sides and complete the parallelogram by drawing lines parallel to those sides.

Step-wise construction (straightedge and compass):

1. Draw segment AB = 4.5 cm.

2. At A, construct angle DAB = 75° (use protractor or compass-and-straightedge angle construction). From A, draw the ray making 75° with AB.

3. On that ray, mark point D so that AD = 5.0 cm.

4. Construct the line through B parallel to AD (copy the direction of AD at point B):

With centre A draw a small arc cutting AB at P and AD at Q.

With the same radius draw an arc with centre B cutting BA (produced if necessary) at R.

Measure the distance PQ with the compass; with centre R and radius PQ mark S.

Join B to S; the ray BS is parallel to AD.

5. Construct the line through D parallel to AB (copy the direction of AB at point D) by the analogous angle-copy procedure:

With centre A (same as step 4) you already have arc points P and Q.

With the same radius draw an arc with centre D to cut the direction where a line parallel to AB would lie; transfer the chord length between P and the intersection on AB to mark the corresponding point on the arc centered at D; join D to that point.

The resulting line through D is parallel to AB.

6. Let C be the intersection of the line through B (parallel to AD) and the line through D (parallel to AB).

7. Join BC and CD to complete the quadrilateral ABCD.

By construction AB || CD and AD || BC and lengths AB = CD = 4.5 cm, AD = BC = 5.0 cm, with ∠A = 75°, so ABCD is a parallelogram with the required data.

ABCD constructed as above is the required parallelogram with AB = 4.5 cm, AD = 5.0 cm and ∠A = 75°.