Question 14

The figure is made of sectors of circles. All sectors have the same radius and a 60-degree central angle.

Step 1 — Choose radius and first segment

Let us choose a suitable radius. Let us call it r. Draw a line segment. Let its length be r. Let us name its endpoints $V_1$ and $V_2$.

Step 2 — Construct the first shaded sector

We will make an equilateral triangle $V_0V_1V_2$. Place the compass point at $V_1$. Open the compass to radius r. Draw an arc above $V_1V_2$. Place the compass point at $V_2$. Open the compass to radius r. Draw another arc above $V_1V_2$. The arcs meet at point $V_0$. Join $V_0$ to $V_1$ with a straight line. Join $V_0$ to $V_2$ with a straight line. Place the compass point at $V_0$. Open the compass to radius r. Draw an arc connecting $V_1$ and $V_2$. This forms the first shaded sector.

Step 3 — Construct the first unshaded sector

We will make an equilateral triangle $V_1V_2V_3$. Place the compass point at $V_1$. Open the compass to radius r. Draw an arc below $V_1V_2$. Place the compass point at $V_2$. Open the compass to radius r. Draw another arc below $V_1V_2$. The arcs meet at point $V_3$. Join $V_2$ to $V_3$ with a straight line. Join $V_1$ to $V_3$ with a straight line. Place the compass point at $V_2$. Open the compass to radius r. Draw an arc connecting $V_1$ and $V_3$. This forms the first unshaded sector.

Step 4 — Construct the second shaded sector

We will make an equilateral triangle $V_2V_3V_4$. Place the compass point at $V_2$. Open the compass to radius r. Draw an arc above $V_2V_3$. Place the compass point at $V_3$. Open the compass to radius r. Draw another arc above $V_2V_3$. The arcs meet at point $V_4$. Join $V_3$ to $V_4$ with a straight line. Join $V_2$ to $V_4$ with a straight line. Place the compass point at $V_3$. Open the compass to radius r. Draw an arc connecting $V_2$ and $V_4$. This forms the second shaded sector.

Step 5 — Construct the second unshaded sector

We will make an equilateral triangle $V_3V_4V_5$. Place the compass point at $V_3$. Open the compass to radius r. Draw an arc below $V_3V_4$. Place the compass point at $V_4$. Open the compass to radius r. Draw another arc below $V_3V_4$. The arcs meet at point $V_5$. Join $V_4$ to $V_5$ with a straight line. Join $V_3$ to $V_5$ with a straight line. Place the compass point at $V_4$. Open the compass to radius r. Draw an arc connecting $V_3$ and $V_5$. This forms the second unshaded sector.

Step 6 — Construct the third shaded sector

We will make an equilateral triangle $V_4V_5V_6$. Place the compass point at $V_4$. Open the compass to radius r. Draw an arc above $V_4V_5$. Place the compass point at $V_5$. Open the compass to radius r. Draw another arc above $V_4V_5$. The arcs meet at point $V_6$. Join $V_5$ to $V_6$ with a straight line. Join $V_4$ to $V_6$ with a straight line. Place the compass point at $V_5$. Open the compass to radius r. Draw an arc connecting $V_4$ and $V_6$. This forms the third shaded sector.

Step 7 — Shade the regions

Shade the first, third, and fifth sectors. These are the sectors $V_0V_1V_2$, $V_3V_2V_4$, and $V_5V_4V_6$. The final figure is now constructed.