Playing with Constructions

Given: The central line AB = 8 cm, and the wave is made of half circles placed along AB.

Concept: A half circle (semicircle) is drawn with its diameter lying on the central line. If the total length AB = 8 cm is divided equally into two half circles, each half circle has a diameter of 4 cm.

Working:
- Each half circle sits on a portion of AB as its diameter.
- Diameter of each half circle = $\frac{8}{2} = 4$ cm
- Radius = $\frac{\text{Diameter}}{2} = \frac{4}{2} = 2$ cm

Answer:
- The radius to be taken in the compass = 2 cm
- The length of AX (the diameter of the first half circle) = 4 cm

Activity-based question — Sample with central line = 12 cm

Given: Let the central line AB = 12 cm.

Steps of Construction:
1. Draw a line segment AB = 12 cm.
2. Divide AB into equal parts. For two waves, mark point X such that AX = 6 cm (midpoint).
3. For the first wave: Place the compass tip at the midpoint of AX (i.e., 3 cm from A), set radius = 3 cm, and draw a semicircle above AB.
4. For the second wave: Place the compass tip at the midpoint of XB (i.e., 3 cm from X), set radius = 3 cm, and draw a semicircle below AB.
5. The wavy wave is complete.

Observation: The radius used = 3 cm, and each half circle has diameter = 6 cm. Students may choose any even length for the central line and divide accordingly to get identical waves.

Given: Waves that are arcs smaller than a semicircle (less than half circles), as seen in the neck of 'A Person'.

Concept: An arc smaller than a semicircle is drawn by placing the compass centre below (or above) the central line, so that the circle's arc crosses the line at two points, producing a shallower curve.

Steps of Construction:
1. Draw the central line AB of a chosen length, say 8 cm.
2. Mark points P and Q on AB such that PQ = 4 cm (the span of one wave). Let P be at 2 cm from A and Q at 6 cm from A.
3. To draw an arc smaller than a semicircle over PQ: Choose a radius larger than half of PQ (i.e., radius $r > 2$ cm, say $r = 3$ cm).
4. Find the centre: It lies on the perpendicular bisector of PQ, below the line AB. Using compass, mark the centre $O_1$ at the intersection of the perpendicular bisector of PQ and a point below AB at the appropriate distance.
5. Place compass tip at $O_1$, set radius = 3 cm, and draw the arc from P to Q above the line.
6. For the second identical wave: Repeat the same process for the next segment of equal length on AB, keeping the same radius and the same offset for the centre.

Key to identical waves: Use the same radius and place the centre at the same perpendicular distance from AB for both waves. This ensures both arcs are congruent.

Note: Multiple trials may be needed to find the right centre position so the arc looks like the one in the figure.