Construct the Fig. 6.6.

Question

Homi · Accepted Answer

Understand the Question

The figure is a wave-like pattern composed of alternating $60^\circ$ circular sectors of equal radius $r$ .
Since each angle in an equilateral triangle is $60^\circ$ and all its sides equal $r$ , the pattern is built by constructing a chain of adjacent equilateral triangles of side length $r$ and drawing circular arcs centered at alternating vertices.
Alternate sectors are then shaded to complete the pattern.

Step 1 · Draw the Base Segment

Choose a suitable radius $r$ . Diagram 1

Draw a line segment $V_1V_2$ of length $r$ .

Step 2 · Construct the First Sector

Construct an equilateral triangle $V_0V_1V_2$ above $V_1V_2$ and draw the bounding arc: Diagram 2

With radius $r$ , draw arcs centered at $V_1$ and $V_2$ above $V_1V_2$ intersecting at $V_0$ .
Join $V_0V_1$ and $V_0V_2$ .
With center $V_0$ and radius $r$ , draw an arc connecting $V_1$ and $V_2$ to form the first sector.

Step 3 · Construct the Second Sector

Construct an equilateral triangle $V_1V_2V_3$ below $V_1V_2$ : Diagram 3

With radius $r$ , draw arcs centered at $V_1$ and $V_2$ below $V_1V_2$ intersecting at $V_3$ .
Join $V_2V_3$ and $V_1V_3$ .
With center $V_2$ and radius $r$ , draw an arc connecting $V_1$ and $V_3$ to form the second sector.

Step 4 · Construct the Third Sector

Construct an equilateral triangle $V_2V_3V_4$ above $V_2V_3$ : Diagram 4

With radius $r$ , draw arcs centered at $V_2$ and $V_3$ above $V_2V_3$ intersecting at $V_4$ .
Join $V_3V_4$ and $V_2V_4$ .
With center $V_3$ and radius $r$ , draw an arc connecting $V_2$ and $V_4$ to form the third sector.

Step 5 · Construct the Fourth Sector

Construct an equilateral triangle $V_3V_4V_5$ below $V_3V_4$ : Diagram 5

With radius $r$ , draw arcs centered at $V_3$ and $V_4$ below $V_3V_4$ intersecting at $V_5$ .
Join $V_4V_5$ and $V_3V_5$ .
With center $V_4$ and radius $r$ , draw an arc connecting $V_3$ and $V_5$ to form the fourth sector.

Step 6 · Construct the Fifth Sector

Construct an equilateral triangle $V_4V_5V_6$ above $V_4V_5$ : Diagram 6

With radius $r$ , draw arcs centered at $V_4$ and $V_5$ above $V_4V_5$ intersecting at $V_6$ .
Join $V_5V_6$ and $V_4V_6$ .
With center $V_5$ and radius $r$ , draw an arc connecting $V_4$ and $V_6$ to form the fifth sector.

Step 7 · Shade Alternate Sectors

Shade the 1st, 3rd, and 5th sectors ( $V_0V_1V_2$ , $V_3V_2V_4$ , and $V_5V_4V_6$ ). Diagram 7

Answer

The figure is constructed by drawing alternating $60^\circ$ sectors of radius $r$ and shading the 1st, 3rd, and 5th sectors.

Common Mistakes

Changing Compass Radius: Altering the compass width during construction will cause the triangles not to be equilateral and the sector arcs to misalign.
Incorrect Arc Center: Placing the compass on the wrong vertex when drawing the circular arc instead of using the apex of each sector.

Question 14