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# Constructing Circles

Constructing Circles

Circles are everywhere around us. A ball, a pizza, a pie, a wheel, a plate, a coin, and so on are a few circular objects that we come across in our daily life. We can construct a circle by tying 2 pencils at their ends together using a string. Have one pencil fixed at the center point and then run the second pencil along. We obtain a circle. Here is the simulation that shows how interestingly, a circle is been constructed.

What do you observe? The distance from the center point to any point on the circle is always the same. In this mini-lesson, let's learn how to construct circles. In the process of learning the construction, let us also learn about circles, points, collinear points, non-collinear points, perpendicular bisector, equidistant, the center of the circle, and the radius of the circle.

## Lesson Plan

 1 How Do You Construct a Circle? 2 Thinking Out of The Box! 3 Important Notes on Constructing Circles 4 Solved Examples on Constructing Circles 5 Interactive Questions on Constructing Circles

## How Do You Construct a Circle?

Before constructing a circle, we need to know about the components of a circle.

### Circle and its Parts

Parts of the circle Explanation
Center The distance from the center of the circle to any point on the circle is the same throughout.
Circumference The distance along the circle is called its circumference
Radius The distance from the center to any point on the circle is the radius of the circle.
Diameter Is the biggest chord and is twice the radius.
Chord A chord is a line segment that connects any two points on the circumference of the circle.
Arc A part of the circumference is called an arc.
Segment The region subtended between an arc and a chord is the segment.
Sector The region subtended between two radii and an arc is the sector.

### Construction of a Circle

• Mark a center point.
• Measure the distance between the tips of the compass and the pencil using the ruler to align with the desired radius.
• Place the metal tip on the center point.
• Keeping the metal tip constant at the center point, rotate the pencil.
• You get a circle!

Thus a circle is constructed.

## How to Construct a Circle Using a Compass?

Through a point in the plane, infinitely many lines can pass. However, through two distinct points in the plane, exactly one line can pass. That is two distinct points uniquely determine a line.

What happens in the case of circles? How many points are at least required to uniquely determine a circle?

It should be obvious that through one point, infinitely many circles can pass.

Even through two points, infinitely many circles can pass.

Let us now see how we can actually construct a unique circle passing through three distinct non-collinear points. The following figure shows three such points, A, B, and C:

The center of that circle must be equidistant from all three points.

This means that we have to locate that point in the plane (call it O) such that OA = OB = OC.

To locate O, recall that any point which is equidistant from two fixed points must lie on the perpendicular bisector of the segment joining those points. Therefore, we proceed as follows:

1. Join the points A and B and draw a perpendicular bisector $$P_1$$ of AB.

Any point on $$P_1$$ will be equidistant from A and B:

2. Join B and C and draw a perpendicular bisector $$P_2$$ of BC.

Any point on $$P_2$$ will be equidistant from B and C:

3. Mark the point of intersection of $$P_1$$ and $$P_2$$ as O.

This point O is equidistant from all of A, B, and C.

Let this be the center of the circle.

4. We draw a circle with O as the center, using the compass with the radius as the measure of length OA (or OB or OC).

Think Tank

1. Can two circles intersect in more than 2 points?

2. Why would two circles that have a common arc, must coincide?

3. How many points can be equidistant from 3 non-collinear points?

### Thus we state the theorem as:

Given any three non-collinear points A, B, and C, there exists a unique circle passing through them.

Find in the simulation how the circle is constructed using 3 distinct non-collinear points.

Important Notes
1. With a center and a given radius, only one circle can be drawn.
2. Given any three non-collinear points A, B, and C, there exists a unique circle passing through them.
3. If A, B, and C are collinear, then the perpendicular bisectors of AB and BC would never intersect (they would be parallel).

## Solved Examples

 Example 1

Charlie wants to construct a circle with a radius of 2 inches. Can you help him construct a circle?

Solution

Locate any point as a center of the circle.
With the help of the ruler, measure the distance between the tip of the compass and the tip of the pencil as 2 inches.

Now with the tip of the compass at the center and 2 inches as the radius, rotate the compass.

You will get a circle of radius of 2 inches.

 Thus a circle of 2-inch radius is constructed.
 Example 2

Sandra marked 2 points on a sheet of paper. She is trying to figure out the number of circles that could be constructed which will pass through the given two points.

Can you help her?

Solution

If there are two points, we can consider them as the endpoints of the diameter to start with.

For the next circle that we try to draw, we let the distance between the two points as the chord to that circle.

By doing so, we will get infinite circles passing through the given two points.

 $$\therefore$$, infinite circles could be constructed if two points are given.

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

The mini-lesson targeted the fascinating concept of constructing circles. The math journey around constructing circles starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

### 1. What are the 5 parts of a circle?

The center of the circle, radius, diameter, circumference and the sector are the 5 parts of a circle.

### 2. What are the tools used to construct a circle?

The compass and the ruler are usually used to construct a circle.

### 3. Can a circle be drawn through 3 points?

Yes, a circle can be drawn through 3 points by creating two chords. The perpendicular bisectors of the chord pass through the center of the circle and a circle could be constructed with that point as a center. Thus a circle could be drawn through 3 points.