Construct the following figure.

Question

Homi · Accepted Answer

The figure is a part of a larger pattern. This larger pattern would consist of 8 identical rhombuses arranged around a central point. Each rhombus would have an angle at the center.

Let us calculate this angle. The total angle in a circle is 360 degrees. The number of rhombuses in the full pattern is 8. The angle for each rhombus is found by dividing the total angle by the number of rhombuses.

$= \frac{360^\circ}{8}$

$\boxed{\mathbf{45^\circ}}$

So, each rhombus in the pattern has an acute angle of 45 degrees. We will construct one such rhombus and then replicate it.

Step 1 — Constructing a 45-degree angle

Let us start by drawing a straight line. Mark a point A on this line. This will be the common vertex for all rhombuses. With A as the center, draw a semicircle. This semicircle cuts the line at two points. Let us call these points B and C. Now, we will construct a 90-degree angle at A. With B as the center, draw an arc above the line. With C as the center and using the same radius, draw another arc that crosses the first arc. Let this intersection point be D. Draw a ray from A through D. This ray is perpendicular to the line BC. So, the angle $\angle DAC$ is 90 degrees. Next, we will bisect this 90-degree angle to get a 45-degree angle. The semicircle drawn earlier also cuts the ray AD. Let us call this intersection point E. With C as the center, draw an arc. With E as the center and using the same radius, draw another arc that crosses the previous arc. Let this intersection point be F. Draw a ray from A through F. This ray bisects $\angle DAC$ . So, the angle $\angle FAC$ is 45 degrees. This is one of the angles we need for our rhombus.

Diagram 1

Step 2 — Constructing one rhombus

We have the angle $\angle FAC = \mathbf{45}$ degrees. This will be an interior angle of our rhombus. Let us choose a convenient length for the side of our rhombus. Let this length be $s$ . With A as the center, draw an arc of radius $s$ . This arc cuts the ray AC at a point. Let us call this point G. The arc also cuts the ray AF at a point. Let us call this point H. So, AG and AH are two sides of our rhombus, and their length is $s$ . Now, we need to find the fourth vertex of the rhombus. Let us call it I. With G as the center, draw an arc of radius $s$ . With H as the center, draw another arc of radius $s$ . These two arcs will intersect at a point. Let us call this point I. Join GI and HI using straight lines. Now, AGIH is one rhombus with all sides equal to $s$ and an angle of $\mathbf{45}$ degrees at A.

Diagram 2

Step 3 — Replicating the rhombus to form the figure

The given figure shows 5 rhombuses arranged side by side. The first, third, and fifth rhombuses are shaded. The second and fourth are unshaded. We have constructed one rhombus AGIH. Carefully erase any extra construction lines or arcs, leaving only the rhombus AGIH. Now, we need to create 5 copies of this rhombus. We can use tracing paper to make a template of the rhombus AGIH. Place the tracing paper over your constructed rhombus and trace its outline. Cut out this traced rhombus. This is your template. Place the template on a fresh sheet of paper. Draw around it to make the first rhombus. Since the first rhombus in the figure is shaded, shade this first rhombus completely. Now, rotate your template by $\mathbf{45}$ degrees around point A (the common vertex). Place the rotated template next to the first rhombus, sharing the side AH (or AG depending on rotation). Draw around the template to make the second rhombus. This rhombus should be left unshaded. Rotate the template again by $\mathbf{45}$ degrees around point A. Draw the third rhombus. Shade this rhombus completely. Continue this process, rotating by $\mathbf{45}$ degrees each time, and alternating between shading and leaving unshaded. Draw a total of 5 rhombuses to match the given figure. The final figure will show 5 rhombuses, with the first, third, and fifth shaded, and the second and fourth unshaded.

Diagram 3

Question 14

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