Chapter 8 Solutions — Playing with Constructions

The Wavy Wave is drawn along a central line AB = 8 cm.

The first wave is a half circle sitting on segment AX.

For the wave to be a half circle, the diameter of the half circle equals AX.

The two waves together span the full length AB = 8 cm, so each wave spans 4 cm.

Therefore: AX = 4 cm.

Radius = AX ÷ 2 = 4 ÷ 2 = 2 cm.

The diagonal of rectangle ABCD: AC = BD = √(AB² + BC²) = √(7² + 4²) = √(49 + 16) = √65 ≈ 8.06 cm.

The farthest distance between X and Y is achieved when X is at one end of AD and Y is at the opposite end of BC.

For example: X = D = (0, 0) and Y = B = (7, 4).

XY = √(7² + 4²) = √65 ≈ 8.06 cm = DB.

Or: X = A = (0, 4) and Y = C = (7, 0).

XY = √(7² + 4²) = √65 ≈ 8.06 cm = AC.

So the farthest distance between X and Y equals the length of the diagonal AC (= BD).

In a rectangle, each angle is 90°. A diagonal divides the angle at each opposite corner into two parts.

For the diagonal to divide each opposite angle into equal parts, each half must be 45°.

In a rectangle with sides a and b, the diagonal makes an angle of arctan(b/a) with one side.

For this angle to be 45°, we need arctan(b/a) = 45°, which means b/a = 1, i.e., a = b.

A rectangle with all sides equal is a square.

Therefore, the rectangle must be constructed as a square (all four sides equal).