Question 19

We will use a compass and a straightedge to construct each figure step-by-step.

Step 1 — Constructing an Inflexed Arc (Figure a)

Let us draw the base and vertical sides first.

1. Draw a horizontal line segment. Let us call its ends A and B.
2. Draw a vertical line upwards from A. Mark a point C on this line.
3. Draw a vertical line upwards from B. Mark a point D on this line. Make sure the length BD is the same as AC.
4. Find the midpoint of the line segment CD. Let us call it M.
5. Draw a vertical line upwards from M. This line is the center line of our figure.
6. Open your compass to the length CM.
7. Place the compass point at C. Draw an arc that starts from C and curves towards the center line.
8. Open your compass to the length DM. This length is the same as CM.
9. Place the compass point at D. Draw an arc that starts from D and curves towards the center line.
10. The two arcs will meet at a point on the center line. Let us call this point E.
11. The figure is formed by the vertical lines AC and BD, and the two arcs CE and DE.

Step 2 — Constructing a Flower-like Shape (Figure b)

This figure can be made using only a compass.

1. Draw a central circle. Let its center be O. Let its radius be R.
2. Mark any point on the circumference of this central circle. Let us call this point A.
3. Place the compass point at A. Keep the compass opening to radius R.
4. Draw a new circle. This circle will pass through O. It will also intersect the central circle at two points. Let us call one of these points B.
5. Place the compass point at B. Keep the compass opening to radius R.
6. Draw another new circle. This circle will intersect the central circle at a new point. Let us call this point C.
7. Continue this process. Use each new intersection point on the central circle as the center for the next circle. Do this until you have drawn six circles around the central one.
8. The central circle and the parts of the six outer circles form the petals.

Step 3 — Constructing a Hexagon inside a Circle (Figure c)

This is a regular hexagon inscribed in a circle.

1. Draw a circle. Let its center be O. Let its radius be R.
2. Mark any point on the circumference of this circle. Let us call this point A.
3. Place the compass point at A. Keep the compass opening to radius R.
4. Draw an arc that intersects the circle at a new point. Let us call this point B.
5. Place the compass point at B. Keep the compass opening to radius R.
6. Draw another arc that intersects the circle at a new point. Let us call this point C.
7. Continue this process. Use each new intersection point on the circle as the center for the next arc. Do this until you have marked six points on the circumference. The last arc should end at point A.
8. Connect these six points with straight lines. This forms the regular hexagon.

Step 4 — Constructing Seven Circles (Figure d)

This figure has a central circle and six circles around it. All circles have the same radius.

1. Draw a central circle. Let its center be O. Let its radius be R.
2. Mark any point on the circumference of this central circle. Let us call this point A.
3. Place the compass point at A. Keep the compass opening to radius R.
4. Draw a new circle. This circle will pass through O. It will also intersect the central circle at two points. Let us call one of these points B.
5. Place the compass point at B. Keep the compass opening to radius R.
6. Draw another new circle. This circle will intersect the central circle at a new point. Let us call this point C.
7. Continue this process. Use each new intersection point on the central circle as the center for the next circle. Do this until you have drawn six circles around the central one.
8. All seven circles (the central one and the six surrounding ones) should be visible.

Step 5 — Constructing a Tiling Pattern (Figure e)

This pattern is a large hexagon made of smaller equilateral triangles and stars.

1. Draw a large circle. Let its center be O. Let its radius be R.
2. Inscribe a regular hexagon inside this circle. Let its vertices be V1, V2, V3, V4, V5, V6. (You can use the method from Step 3).
3. Draw straight lines from the center O to each vertex V1, V2, ..., V6. This divides the large hexagon into six large equilateral triangles. For example, triangle OV1V2.
4. Now, let us work on one of these large equilateral triangles, for example, OV1V2. a. Divide each side of this triangle (OV1, OV2, and V1V2) into three equal parts. Mark these division points. b. Draw lines connecting these division points. Make sure these lines are parallel to the sides of the triangle. This will divide the large triangle OV1V2 into nine smaller equilateral triangles.
5. Repeat step 4 for all six large equilateral triangles.
6. The complete pattern will emerge from these lines, forming the stars and smaller hexagons.