Practice: Solved Examples

Example 1: The Wavy Line

Problem: Construct a wave pattern on a line segment $AB = 12$ cm consisting of 3 equal semicircles.

Solution:

1. Analyze: Total length is 12 cm. There are 3 semicircles.
   • Diameter of each semicircle $= 12 \div 3 = 4$ cm.
   • Radius of each semicircle $= 2$ cm.
2. Mark Points:
   • Draw $AB = 12$ cm.
   • Mark points $P$ and $Q$ at 4 cm and 8 cm from $A$.
   • The segments are $AP$, $PQ$, and $QB$.
3. Find Centers:
   • Midpoint of $AP$ is $C_1$ (at 2 cm).
   • Midpoint of $PQ$ is $C_2$ (at 6 cm).
   • Midpoint of $QB$ is $C_3$ (at 10 cm).
4. Draw Arcs:
   • Place compass at $C_1$, radius 2 cm. Draw semicircle Up.
   • Place compass at $C_2$, radius 2 cm. Draw semicircle Down.
   • Place compass at $C_3$, radius 2 cm. Draw semicircle Up.

Example 2: The Rhombus

Problem: Construct a shape with 4 sides of length 5 cm, but diagonal length 8 cm.

Solution:

1. Draw Diagonal: Draw line segment $AC = 8$ cm.
2. Locate Vertices: We need points $B$ and $D$ such that they are 5 cm away from both $A$ and $C$.
3. Upper Point (B):
   • Center $A$, radius 5 cm $\to$ Draw arc above $AC$.
   • Center $C$, radius 5 cm $\to$ Draw arc intersecting the first one. Mark as $B$.
4. Lower Point (D):
   • Center $A$, radius 5 cm $\to$ Draw arc below $AC$.
   • Center $C$, radius 5 cm $\to$ Draw arc intersecting the first one. Mark as $D$.
5. Connect: Join $AB, BC, CD, DA$.
6. Verify: This is a Rhombus (all sides 5 cm) but not a square (diagonal is 8 cm, not $5\sqrt{2} \approx 7.07$ cm).