Question 8

We will construct the 8-petalled figure by first drawing the central framework of 8 equally spaced lines and then drawing the petals using these lines and a circle.

Step 1 — Draw a central cross

Let us start by drawing a straight line. Let us call this line AB.

Let us find the midpoint of AB. We will call this point O. To do this, we open our compass to a radius more than half of AB. With A as the center, we draw an arc above and below AB. With B as the center, we draw another arc above and below AB. These arcs will intersect at two points. Let us call them C and D. We join C and D with a straight line. The line CD will cut AB exactly in half at point O. The line CD is also perpendicular to AB. So, we now have two perpendicular lines passing through O.

Step 2 — Create 8 radial lines

We now have four lines radiating from O (OA, OB, OC, OD). These lines are $90^\circ$ apart. We need 8 lines, so we need to bisect the angles between these lines. Let us bisect the angle BOC. With O as the center, we draw an arc that cuts OB and OC. Let these points be E and F. With E as the center, we draw an arc. With F as the center, we draw another arc with the same radius. These two arcs intersect at a point. Let us call it G. We draw a line from O through G. This line bisects angle BOC. We repeat this process for angles COA, AOD, and DOB. After bisecting all four $90^\circ$ angles, we will have 8 lines radiating from O. Each line will be $45^\circ$ apart from its neighbours.

Step 3 — Mark petal tips

Let us choose a suitable radius for our petals. Let this radius be R. With O as the center, we draw a circle with radius R. This circle will intersect our 8 radial lines at 8 points. Let us mark these 8 points. These points will be the tips of our petals. Let us call them $P_1, P_2, ..., P_8$.

Step 4 — Draw the petals

Now we will draw the curved sides of each petal. Each petal will connect the center O to one of the points $P_i$. Let us focus on one petal, for example, the one connecting O to $P_1$. We need to find the centers for the two arcs that form this petal. First, we draw the perpendicular bisector of the line segment $OP_1$. To do this, we open our compass to a radius more than half of $OP_1$. With O as the center, we draw an arc. With $P_1$ as the center, we draw another arc with the same radius. These arcs intersect at two points. We join these points to get the perpendicular bisector of $OP_1$. Now, we find where this perpendicular bisector cuts the two radial lines adjacent to $OP_1$. Let these adjacent radial lines be $OP_8$ and $OP_2$. Let the perpendicular bisector of $OP_1$ intersect line $OP_8$ at point $C_1$. Let it intersect line $OP_2$ at point $C_2$. Now, with $C_1$ as the center and radius $C_1O$, we draw an arc from O to $P_1$. With $C_2$ as the center and radius $C_2O$, we draw another arc from O to $P_1$. These two arcs form one petal. We repeat this process for all 8 petals. Each petal will have its own pair of arc centers.

Step 5 — Final figure

After drawing all 8 petals, we erase all the construction lines and arcs (like AB, CD, angle bisectors, the circle, and the perpendicular bisectors). We are left with the beautiful 8-petalled figure.

Answer

Draw a line segment AB. Construct its perpendicular bisector CD intersecting at O. Bisect the four $90^\circ$ angles around O to create 8 equally spaced radial lines. Draw a circle with center O and a chosen radius R. Mark the 8 intersection points as $P_1, ..., P_8$. For each petal (e.g., from O to $P_1$): Construct the perpendicular bisector of $OP_1$. Find where this bisector intersects the two adjacent radial lines (e.g., $OP_8$ and $OP_2$). Let these points be $C_1$ and $C_2$. Draw an arc with center $C_1$ and radius $C_1O$ from O to $P_1$. Draw an arc with center $C_2$ and radius $C_2O$ from O to $P_1$. Repeat for all 8 petals. Erase all construction lines to reveal the 8-petalled figure.