Playing with Constructions | FIO

Question 3

Try to recreate the figure where the waves are smaller than a half circle (as appearing in the neck of the figure, 'A Person'). The challenge here is to get both the waves to be identical. This may be tricky!

Solution

Understand the Question

To recreate the figure 'A Person', we construct a geometric drawing consisting of:
- A rectangular body of dimensions $6\text{ cm} \times 5\text{ cm}$ .
- A circular head of radius $2\text{ cm}$ centered above the midpoint of the body's top edge.
- Two identical circular arcs (waves) of radius $2\text{ cm}$ connecting the shoulders to the neck midpoint.
Because the radius ( $2\text{ cm}$ ) is greater than half the span of each wave ( $1.5\text{ cm}$ ), the resulting arcs are strictly smaller than a semicircle.

Step 1 · Draw the Body

Draw a rectangle $ABCD$ where:

Width $AB = CD = 6\text{ cm}$
Height $AC = BD = 5\text{ cm}$
$C$ and $D$ are the top-left and top-right vertices, while $A$ and $B$ are the bottom-left and bottom-right vertices.

Step 2 · Mark the Neck Midpoint

Find and mark the midpoint $E$ of the top side $CD$ :

$CE = ED = 3\text{ cm}$ Diagram 2

Step 3 · Draw the Neck and Head

From point $E$ , draw a vertical line segment upwards of length $2\text{ cm}$ and mark the endpoint as $F$ .
With $F$ as the center, draw a circle of radius $2\text{ cm}$ to form the head.

Step 4 · Construct the Identical Neck Waves

To construct two identical arcs on segments $CE$ and $ED$ :

Set compass radius to $2\text{ cm}$ (which is greater than $\frac{1}{2}CE = 1.5\text{ cm}$ ).
First arc ( $C$ to $E$ ):
- With compass at $C$ , draw an arc above line $CE$ .
- With compass at $E$ , draw another arc intersecting the first at point $O_1$ .
- With center $O_1$ and radius $2\text{ cm}$ , draw arc $\overparen{CE}$ .
Second arc ( $E$ to $D$ ):
- With compass at $E$ , draw an arc above line $ED$ .
- With compass at $D$ , draw another arc intersecting the first at point $O_2$ .
- With center $O_2$ and radius $2\text{ cm}$ , draw arc $\overparen{ED}$ .

Since the radius $r = 2\text{ cm} > 1.5\text{ cm}$ , both arcs are identical and smaller than a semicircle. Diagram 4

Answer

The figure is constructed with a rectangular body ( $6\text{ cm} \times 5\text{ cm}$ ), a circular head of radius $2\text{ cm}$ , and two identical neck arcs of radius $2\text{ cm}$ .

Common Mistakes

Radius Too Small: Setting the compass width to less than $1.5\text{ cm}$ (half of $CE$ ), which prevents the intersecting arcs from meeting to find centers $O_1$ and $O_2$ .
Non-Identical Waves: Changing the compass radius between constructing the first arc $\overparen{CE}$ and the second arc $\overparen{ED}$ .
Drawing Semicircles: Using a radius of $1.5\text{ cm}$ with centers directly on line $CD$ , which produces exact semicircles instead of arcs smaller than a half circle.

Question 3

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