Construct the 8-petalled figure shown in Fig. 6.5.

Question

Homi · Accepted Answer

Understand the Question

To construct an 8-petalled geometric flower pattern:

First, establish an 8-ray coordinate framework by constructing two perpendicular lines intersecting at a center point $O$ , then bisecting the four $90^\circ$ angles to obtain eight equally spaced radial lines ( $45^\circ$ apart).
Next, draw a circle centered at $O$ to mark 8 petal tips ( $P_1, P_2, \dots, P_8$ ) on the rays.
Finally, construct circular arcs connecting the center $O$ to each tip $P_i$ using centers found on adjacent radial lines, completing the symmetrical 8-petalled design.

Step 1 · Draw Central Perpendicular Lines

Draw a straight line segment $\text{AB}$ . Diagram 1

With a compass radius greater than half of $\text{AB}$ , draw arcs from centers $\text{A}$ and $\text{B}$ above and below $\text{AB}$ .
Mark the intersection points of these arcs as $\text{C}$ and $\text{D}$ .
Join $\text{CD}$ to intersect $\text{AB}$ at point $\text{O}$ . Line $\text{CD} \perp \text{AB}$ at center $\text{O}$ .

Step 2 · Construct 8 Radial Lines

The rays $\text{OA, OB, OC, OD}$ form four $90^\circ$ angles around $\text{O}$ . Diagram 2

Bisect $\angle \text{BOC}$ : With center $\text{O}$ , draw an arc cutting $\text{OB}$ at $\text{E}$ and $\text{OC}$ at $\text{F}$ . From $\text{E}$ and $\text{F}$ , draw equal intersecting arcs to find point $\text{G}$ . Draw ray $\text{OG}$ .
Repeat the bisection for $\angle \text{COA}$ , $\angle \text{AOD}$ , and $\angle \text{DOB}$ .

This yields 8 radial lines around $\text{O}$ , each spaced $45^\circ$ apart.

Step 3 · Mark the Petal Tips

Choose a radius $R$ for the petals. Diagram 3

With center $\text{O}$ and radius $R$ , draw a circle. Mark the 8 intersection points on the radial lines as $P_1, P_2, \dots, P_8$ .

Step 4 · Construct the Petal Arcs

For each petal (e.g., from $\text{O}$ to $P_1$ ): Diagram 4

Construct the perpendicular bisector of segment $\text{O}P_1$ .
Let this bisector intersect the two adjacent radial lines $\text{O}P_8$ and $\text{O}P_2$ at points $C_1$ and $C_2$ respectively.
With center $C_1$ and radius $C_1\text{O}$ , draw an arc from $\text{O}$ to $P_1$ .
With center $C_2$ and radius $C_2\text{O}$ , draw an arc from $\text{O}$ to $P_1$ .

Repeat this procedure for all 8 petals.

Step 5 · Complete the Figure

Erase all temporary construction lines, rays, bisectors, and the outer circle to leave only the symmetrical 8-petalled flower outline. Diagram 5

Answer

The 8-petalled figure is constructed by creating 8 radial lines at $45^\circ$ intervals, marking tips $P_1, \dots, P_8$ at radius $R$ , and drawing circular arcs connecting center $\text{O}$ to each tip using centers located on adjacent radial lines.

Common Mistakes

Unequal Angular Spacing: Inaccurate angle bisection will make petals uneven in size and asymmetric.
Incorrect Arc Centers: The centers for petal arcs must lie exactly on the adjacent radial lines where the perpendicular bisector of $\text{O}P_i$ intersects them, not chosen arbitrarily.
Compass Radius Drift: Changing the compass radius while drawing corresponding arcs will distort the symmetry of the petals.

Question 8

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