Question

Draw a rectangle of area 18 square centimetres and a square of same area.

Hint:

The geometric mean construction is a powerful tool for converting a rectangle of sides a and b into a square of equal area. The side of the square will be the length of the perpendicular from the junction of a and b to a semicircle drawn on a+b as a diameter.

Answer 1

The task is to construct two shapes: a rectangle with an area of 18 cm², and then a square with an equal area. This involves constructing a side of length √(18) cm.

Rectangle:
We need two numbers that multiply to 18. A simple choice is length = 6 cm and width = 3 cm.

Square:
The area is 18 cm², so the side length is √(18) cm. We can construct this length using the geometric mean theorem: the side of the square is the geometric mean of the rectangle's sides (√(6 × 3)).

Part 1: Construct the Rectangle (6 cm x 3 cm)
1. Draw a line segment AB = 6 cm.
2. At point B, use a protractor or compass to construct a 90° angle.
3. Along the perpendicular line, measure and mark point C such that BC = 3 cm.
4. With C as the center and radius 6 cm, draw an arc. With A as the center and radius 3 cm, draw another arc to intersect the first one at point D.
5. Join AD and CD. ABCD is the required rectangle.

Part 2: Construct the Square (Area 18 cm²)
1. Extend the side AB of the rectangle to a point E such that BE = BC = 3 cm. The total length of AE is now 6+3=9 cm.
2. Find the midpoint of the segment AE by constructing its perpendicular bisector. Let the midpoint be M. 3. With M as the center and MA as the radius, draw a semicircle on AE.
4. At point B, construct a line perpendicular to AE. Let it intersect the semicircle at point F.
5. The length of the segment BF is √(AB × BE) = √(6 × 3) = √(18) cm.
6. Now, construct a square with side length equal to BF. Draw a base PQ = BF. Construct perpendiculars at P and Q, and mark points R and S such that QR = PS = PQ. Join RS to complete the square PQRS.