Playing with Constructions | A

Question 2

Mark a point $P$ in your notebook. Then, mark as many points as possible, in different directions, that are $4\text{ cm}$ away from $P$ .

Think: Imagine marking all the points of $4\text{ cm}$ distance from the point $P$ . How would they look?

Try to draw it and verify if it is correct by taking some points on the curve and checking if their distances from $P$ are indeed $4\text{ cm}$ . Explore, if you have not already done so, and see if a compass can be used for this purpose.

You can start by marking a few points of distance $4\text{ cm}$ from $P$ using the compass. How can this be done?

Solution

Understand the Question

A circle is defined as the collection of all points in a plane that are at a fixed distance from a fixed point.
The fixed central point is the center ( $P$ ), and the fixed distance is the radius ( $4\text{ cm}$ ).
By marking points $4\text{ cm}$ away from $P$ in all possible directions and connecting them, they form a continuous circular boundary.

Step 1 · Mark the Center Point

Mark a fixed reference point in your notebook and label it $P$ . Diagram 1

Step 2 · Mark Points at Equal Distance in Different Directions

Using a ruler, measure and mark several points that are each at a distance of $4\text{ cm}$ from $P$ :

Point $Q$ , $4\text{ cm}$ to the right of $P$
Point $R$ , $4\text{ cm}$ above $P$
Point $S$ , $4\text{ cm}$ to the left of $P$
Point $T$ , $4\text{ cm}$ below $P$

Marking more points in all possible directions begins to outline a continuous, smooth closed curve.

Step 3 · Draw the Complete Circle Using a Compass

To draw all possible points simultaneously:

Open a pair of compasses to a radius of $4\text{ cm}$ using a ruler.
Place the metallic needle tip firmly on point $P$ .
Rotate the pencil tip $360^\circ$ around point $P$ .

The traced curve forms a circle where every point on its boundary is exactly $4\text{ cm}$ away from point $P$ .

Answer

The collection of all points at a distance of $4\text{ cm}$ from $P$ forms a circle with center $P$ and radius $4\text{ cm}$ .

Common Mistakes

Discontinuous Points: Thinking that only a few discrete points exist at $4\text{ cm}$ , rather than recognizing that infinitely many points form a continuous closed curve (circle).
Compass Width Shift: Allowing the compass opening to shift from $4\text{ cm}$ while rotating, which causes an irregular oval instead of a true circle.

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