Question

Construct a parallelogram ABCD in which AB = 5 cm, AC = 6.4 cm and height corresponding to AB is 4 cm.

Answer 1

Given: AB = 5 cm, AC = 6.4 cm and the altitude (height) to side AB = 4 cm.

Step-wise calculation:

1. Put AB on a horizontal line with A at (0, 0) and B at (5, 0).

2. Because the height to AB is 4 cm, point C must lie on the line y = 4 the line parallel to AB at distance 4. Write C = (x, 4).

3. Use AC = 6.4 cm:

AC² = x² + 4² 

= x² + 16 

= 6.4²

= 40.96 

⇒ x² = 40.96 – 16

= 24.96 

⇒ `x = ±sqrt(24.96)`

= ± 4.996 

So, there are two possible intersection points:

C₁ = (+4.996, 4)

C₂ = (–4.996, 4)

4. Compute the corresponding side AD which equals BC:

For C₁: `BC = sqrt((4.996 - 5)^2 + 4^2)`

= `sqrt(0.000016 + 16)` 

= 4.000002

= 4.00 cm

So, AD = 4 cm.

For C₂: `BC = sqrt((-4.996 - 5)^2 + 4^2)`

= `sqrt(99.920 + 16)`

= `sqrt(115.92)`

= 10.77 cm

So, AD = 10.77 cm.

5. Area check: Area = Base × Height 

= AB × Height

= 5 × 4

= 20 cm²

Construction (compass and straightedge):

1. Draw segment AB = 5 cm.

2. Construct a line l parallel to AB at distance 4 cm e.g., at A erect a perpendicular to AB, measure 4 cm along it, then through that point draw a line parallel to AB. Line 1 is the locus where C must lie.

3. With center A and radius 6.4 cm draw a circle. Mark its intersection(s) with line l; these intersection point(s) are the possible C (two possible C as computed above).

4. For a chosen intersection C:

Draw through C a line parallel to AB.

Draw through A a line parallel to BC. The intersection of these two lines is D.

5. Join A–B–C–D to complete parallelogram ABCD.