Question

Construct a rectangle ABCD in which AB = 5.2 cm and BD = 6.5 cm.

Answer 1

Given: Rectangle ABCD with AB = 5.2 cm and diagonal BD = 6.5 cm.

Step-wise calculation:

1. Use the Pythagorean relation in right triangle ABD (AB ⟂ BC in a rectangle):

BD² = AB² + BC²

2. Compute BC:

BD² = 6.5²

= 42.25

AB² = 5.2²

= 27.04

BC² = BD² – AB²

= 42.25 – 27.04

= 15.21

`BC = sqrt(15.21)`

= 3.9 cm

3. Construction steps (compass and straightedge):

Draw segment AB = 5.2 cm.

At point B construct a line perpendicular to AB.

On this perpendicular, mark point C so that BC = 3.9 cm.

Through C draw a line parallel to AB.

Through A draw a line parallel to BC or construct the perpendicular to AB at A; the intersection of these two lines is D.

Join AD and CD.

Verify: `BD = sqrt(AB^2 + BC^2)`

= `sqrt(27.04 + 15.21)`

= `sqrt(42.25)`

= 6.5 cm

Rectangle ABCD with AB = 5.2 cm, BC = 3.9 cm and diagonal BD = 6.5 cm is constructed; all checks satisfy BD² = AB² + BC², so the figure is the required rectangle.