Construct the following figure.

How do we construct the petals so that they are of the maximum possible size within a given square?

Question

Construct the following figure. How do we construct the petals so that they are of the maximum possible size within a given square?

Homi · Accepted Answer

Understand the Question

To construct four petals of the maximum size inside a square, we start by drawing a square $ABCD$ .
Next, locate the midpoints $P, Q, R, S$ on the four sides of the square.
Using the corners of the square as centers, draw circular arcs between adjacent midpoints to form each symmetrical petal.

Step 1 · Draw the Square

Diagram 1

Draw a line segment $AB$ of the chosen side length.
At point $A$ , construct a perpendicular line segment $AD$ such that $AD = AB$ .
At point $B$ , construct a perpendicular line segment $BC$ such that $BC = AB$ .
Join $C$ and $D$ to complete the square $ABCD$ .

Step 2 · Find the Midpoints of the Sides

Diagram 2

Use a compass to find the perpendicular bisector of each side:
- Midpoint of $AB$ : Point $P$
- Midpoint of $BC$ : Point $Q$
- Midpoint of $CD$ : Point $R$
- Midpoint of $DA$ : Point $S$
These four midpoints mark where the petals meet the square.

Step 3 · Draw the Arcs to Form Petals

Diagram 3

Draw arcs using the square vertices as centers:

Top-left petal (near $A$ ):
- Center at $D$ , radius $DP$ : Draw arc from $P$ to $S$ .
- Center at $B$ , radius $BS$ : Draw arc from $S$ to $P$ .
Top-right petal (near $B$ ):
- Center at $A$ , radius $AQ$ : Draw arc from $P$ to $Q$ .
- Center at $C$ , radius $CP$ : Draw arc from $Q$ to $P$ .
Bottom-right petal (near $C$ ):
- Center at $B$ , radius $BR$ : Draw arc from $Q$ to $R$ .
- Center at $D$ , radius $DQ$ : Draw arc from $R$ to $Q$ .
Bottom-left petal (near $D$ ):
- Center at $C$ , radius $CS$ : Draw arc from $R$ to $S$ .
- Center at $A$ , radius $AR$ : Draw arc from $S$ to $R$ .

Answer

The four petals of maximum size are constructed by drawing circular arcs connecting adjacent side midpoints ( $P, Q, R, S$ ) using the vertices of the square as centers.

Common Mistakes

Center Confusion: Placing the compass needle at the side midpoints rather than the square's corners ( $A, B, C, D$ ) to draw the arcs.
Incorrect Radius: Not adjusting the compass radius to precisely touch the opposite side midpoints, which leads to asymmetrical petals.

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