Constructions Based on Circle: Explanation, Steps, Examples

Constructions Based on Circle: A circle is a closed curve formed by uniting all points in a plane at a constant distance from a fixed point in the same plane. There are circles everywhere around us. A ball, a pizza, a pie, a wheel, a plate, a coin, and so on are examples of circular items we encounter daily. We can make a circle by connecting the ends of two pencils with a string. Place one pencil in the centre and then run the second pencil along the line. We end up with a circle. 

In this topic, let us learn constructions about circumcircle, incircle, and tangents to a circle.

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Construction of Circles

A circle is defined as the path followed by a point moving so that its distance from a fixed point remains constant. The constant distance between the fixed point and the circle’s circumference is known as the radius of the circle. As a result, a circle can be drawn if the radius and centre of a circle are known.

Example: Make a circle with a radius of \(2.5 \mathrm{~cm}\).

Steps of construction:

Step 1: On a sheet of paper, make a point \(O\) where a circle will be drawn.

Step 2: Using a scale, measure \(2.5 \mathrm{~cm}\) using a set of compasses.

Step 3: Keep the needle at mark \(O\) and draw a complete arc holding the compasses from its knob without disturbing the opening of the compasses. We get the required circle after completing one complete round.

Constructing Circumcircle of a Triangle

The circumscribed circle is a circle that passes through all the vertices of a polygon. The circumscribed circle’s centre is called the circumcentre, and the polygon is referred to as the inscribed polygon.

To construct a circle circumscribing a given triangle (say, \(\Delta ABC)\), draw the perpendicular bisectors of any two sides of the triangle. Let these perpendicular bisectors meet at point \(O\). Taking point \(O\) as centre and \(OA\) or \(OB\) or \(OC\) as radius, draw a circle that will pass through all the three vertices of the triangle. Here, point \(O\) is called the circumcentre of the triangle and \(OA=OB=OC\) is called its circumradius.

To construct a circumscribing circle of a triangle. Let \(ABC\) be the given triangle. 

Steps: 

1. Draw the perpendicular bisectors of any two sides of the triangle. Let the perpendicular bisectors of \(AB\) and \(AC\) be drawn, which meet at point \(O.\)
2. Taking \(O\) as the centre and radius equal to \(OA\) (or, \(OB\) or, \(OC)\), draw a circle. 

The circle so obtained is the required circle.

Constructing Incircle of a Triangle

To construct an inscribed circle in each triangle (say, \(\Delta ABC\)), draw the bisectors of any two angles of the triangle. Let these angle bisectors meet at point \(I.\) From the point \(I,\) draw \(ID\) perpendicular to any side of the given \(\Delta ABC\). Now with I as centre and \(ID\) as radius, draw a circle that will touch all the three sides of the given \(\Delta ABC\). Here, the point \(I\) is called the incentre of the triangle and \(ID\) is called inradius. Whether the triangle is regular (equilateral triangle) or not, the methods for the above constructions will be the same.

To construct an inscribed circle of a triangle. Let \(ABC\) be the given triangle.

Steps:

1. Draw the bisectors of any two angles of the triangle. Let the bisectors of angles \(A\) and \(B\) are drawn, and they meet at \(I.\)
2. From \(I,\) drop perpendicular to any side of the triangle. Let \(ID\) be the perpendicular drawn from \(I\) to side \(BC\). 
3. With \(I\) as centre and \(ID\) as radius, draw a circle that will touch all the three sides of the triangle. The circle so obtained is the required incircle.

What is Tangent to a Circle?

The line which touches a circle at one point only is called the tangent of the circle. In the given figure, \(EF\) is tangent to the given circle at point \(P\) of the circle. The point at which a tangent touches the circle is called the point of contact.

To construct a tangent to a given circle through a point on its circumference. 

Let the centre of the given circle be \(O,\) and \(P\) be any point on its circumference. 

Steps: 

1. Join \(O\) and \(P\). 
2. Draw line \(APB\) making an angle of \(90^{\circ}\) with \(OP\), i.e. \(\angle O P A=90^{\circ}\)

\(APB\) is the required tangent to the given circle through a point \(P\) on its circumference.

Note: The angle between the radius and the tangent at the point of contact is \(90^{\circ} .\)

Constructing a Tangent to a Circle from an External Point

Draw a tangent to a circle of radius \(1.5 \mathrm{~cm}\) from a point at a distance \(5 \mathrm{~cm}\) from the centre.

Steps of Construction: 

1. First, draw a circle with a centre \(O\) and radius \(1.5 \mathrm{~cm}\)
2. Take a point \(P\) at a distance \(5 \mathrm{~cm}\) from the centre. Join \(O\) and \(P.\) Then the line segment is \(OP.\) 
3. Bisect the line segment \(OP\) at \(T.\)
4. Taking \(T\) as centre and \(OT\) as radius draw a semicircle which passes through \(P\). This semicircle meets the circle with centre \(O\) at \(Q.\)
5. Join \(P\) and \(Q\) by a line segment and produce it. Then \(PQ\) is the tangent from an exterior point \(P\) to the circle with centre \(O.\)

Construction of Two Tangents from a Point Outside of the Circle 

To construct tangents to a given circle from an exterior point. Let the centre of the given circle be \(O\) and \(P\) be an exterior point, i.e., \(P\) lies outside the circle. 

Steps: 

1. Join \(P\) and \(O.\) 
2. Draw a circle with \(OP\) as diameter, which cuts the given circle at points \(A\) and \(B.\) 
3. Join \(PA\) and \(PB. PA\) and \(PB\) are the required tangents to the given circle from an exterior point \(P.\)

Note: Tangents drawn to a circle from an exterior point are always equal in length, i.e. \(PA=PB.\)

Solved Examples – Constructions Based on Circle

Q.1. Draw a circle of radius \(3.5\,{\rm{cm}}\) Mark a point \(P\) outside the circle at a distance of \(6\,{\rm{cm}}\) from the centre. Construct two tangents from \(P\) to the given circle. Measure and write down the length of one tangent.
Ans: Steps of construction:
1. Mark a point \(O.\) With \(O\) as centre and radius \(3.5 \mathrm{~cm}\), draw a circle. 
2. Take a point \(P\) at \(6 \mathrm{~cm}\) from \(0. P\) lies outside the circle. 
3. Join \(OP\) and draw its perpendicular bisector to meet \(OP\) at \(M.\) 
4. Taking \(M\) as centre and \(OM\) (or \(MP)\) as radius, draw a circle. Let this circle intersect the previous circle at points \(A\) and \(B.\) 
5. Join \(PA\) and \(PB\). Then \(PA\) and \(PB\) are the required tangents. On measuring, we find that \(A P=B P=4.9 \mathrm{~cm}\) (approximately).

Justification:
Join \(OA\), then \(\angle O A P=90^{\circ}\) (angle in a semicircle \(\left.=90^{\circ}\right)\)
As \(OA\) is radius and \(\angle O A P=90^{\circ}\), so \(PA\) must be tangent to the circle.

Q.2. Draw two concentric circles of radii \(3\,{\rm{cm}}\) and \(5\,{\rm{cm}}.\) Taking a point on the outer circle, construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculations. 
Ans: Steps of construction:
1. Draw two concentric circles of radii \(3 \mathrm{~cm}\) and \(5 \mathrm{~cm}\) with point \(O\) as their centre. 
2. Let \(P\) be a point on the outer circle. Join \(OP\) and draw its perpendicular bisector to meet \(OP\) at \(M.\) 
3. Taking \(M\)  as a centre and \(OM\)  (or \(MP)\)  as radius, draw a circle. Let this circle intersect the smaller circle, i.e. Circle of radius \(3 \mathrm{~cm}\) at points \(A\)  and \(B.\)  
4. Join \(PA\)  and \(PB\). Then \(PA\) and \(PB\) are the required tangents on measuring \(PA\)  (or \(PB)\); we find that \(P A=4 \mathrm{~cm}.\)

Calculation of length \(PA\)
Join \(OA.\)
In \(\Delta OAP,\angle OAP = {90^ \circ }\) (angle in a semicircle)
By Pythagoras theorem, we get
\(P A^{2}=O P^{2}-O A^{2}=5^{2}-3^{2}=25-9=16\)
\(\Rightarrow P A=4 \mathrm{~cm}\)

Q.3. Construct the circumscribed circle of a triangle with sides \(AB = 4.5\,{\rm{cm}},\,BC = 4\,{\rm{cm}}\) and \(CA = 3.5\,{\rm{cm}}{\rm{.}}\)
Ans: To construct the circumcircle of \(\Delta ABC\)
Steps of construction.
Construct \(\Delta ABC\) with the given data. 
Draw the perpendicular bisectors of \(AB\) and \(BC\). Let these bisectors meet at the point \(O.\)

3. With \(O\) as centre and radius equal to \(OA\), draw a circle. The circle so drawn passes through the points \(A, B\) and \(C\), and is the required circumcircle of \(\Delta ABC\)

Q.4. Construct a ABC, given that \(AB = 4.5\,{\rm{cm,}}\,BC = 7\,{\rm{cm}}\) and median \(AD = 4\,{\rm{cm}}.\)
Construct an inscribed circle of \(ABC\) and measure its radius. 
Ans: Steps of construction
1. Draw a line segment \(BC\) of length \(7 \mathrm{~cm}\)
2. Draw right bisector of \(BC\) to meet \(BC\) at \(D.\) 
3. Taking \(B\) as a centre of radius \(4.5 \mathrm{~cm}\), draw an arc. Taking \(D\) as centre and radius \(4 \mathrm{~cm}\), draw an arc to meet the previous arc at \(A.\) 
4. Join \(AB\) and \(AC\), then \(ABC\) is the required triangle with the given data. 
5. Draw internal bisector of \(\angle A B C\) and \(\angle A C B\). Let these bisectors of angles meet at \(I.\)
6. From \(I\), draw \(IN\) perpendicular to the side \(BC\). 
7. With \(I\) as centre and radius equal to \(IN\), draw a circle. The Circle so drawn touches all the sides of \(\Delta ABC\), and is the required incircle of \(\Delta ABC\)

On measuring, we find that \(I N=1.5 \mathrm{~cm}\) (approx.)
The radius of incircle \(=1.5 \mathrm{~cm}\) (approx.)

Q.5. Draw a line segment \(AB\) of length \(5\,{\rm{cm}}.\) Make a point \(C\) on \(AB\) such that \(AC = 3\,{\rm{cm}}.\) Using a ruler and compass only, construct:
i. A circle of a radius \({\rm{2}}{\rm{.5}}\,{\rm{cm}},\) passing through \(A\) and \(C.\) 
ii. Construct two tangents to the Circle from the external point \(B.\) Measure and record the length of the tangents.
Ans: Steps of construction
1. Draw a line segment \(AB\) of length \(5 \mathrm{~cm}\). Mark a point \(C\) on \(AB\) such that \(A C=3 \mathrm{~cm}\)
2. Draw the right bisector of \(AC.\) 
3. Taking \(A\) as centre and radius \(2.5 \mathrm{~cm}\) draw an arc to meet the right bisector of \(AC\) at \(0.\) 
4. Taking \(O\) as centre and radius \(OA\), draw a circle. It is the required Circle of radius \(2.5 \mathrm{~cm}\) and passing through points \(A\) and \(C.\) 
5. Join \(OB\) and draw its right bisector to meet \(OB\) at \(M.\) Taking \(M\) as centre and \(OM\) as radius, draw a circle to meet the previous circle at point \(P\) and \(Q.\)

Summary 

In this article, we learnt about the construction of circles. We constructed circumcircle of a triangle, constructing incircle of a triangle, the definition of the tangent to a circle, constructing a tangent to a circle from an external point, construction of two tangents from a point outside of the circle, solved examples on constructions based on circle and FAQs on constructions based on the circle.

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Frequently Asked Questions (FAQs)

Q.1. What are the 4 basic constructions in geometry?
Ans: The four different types of construction are:
1. Constructing line and angle bisectors.
2. Constructing regular polygons inscribed by circles.
3. Constructing incircles and circumcircles.
4. Constructing a tangent line to a circle.

Q.2. How do you perform constructions related to circles?
Ans: Connect the point to the circle’s centre by drawing a line. Construct the line’s perpendicular bisector. Place the compass on the centre of the circle, adjust the length to reach the endpoint, then draw an arc across it.

Q.3. What is basic geometric construction?
Ans: The basic geometric constructions are drawing lines, angles, and other geometric shapes and figures with only a compass and a straightedge (often a ruler without measurements) and no specific length, angle, or other measurements. It’s also possible to copy line segments with it.

Q.4. Which geometric tool is used to draw a circle?
Ans: To draw circles, a compass is used. Its design is like that of a divider, except that the compass has a place for attaching a pencil or lead to one of the compass’s legs.

Q.5. What is circle construction?
Ans: Steps to construct a circle:
1. Mark the centre of the circle.
2. Take the ruler and measure the radius.
3. Draw the circle by placing the compass on the centre point and draw a complete arc.

We hope this detailed article on constructions based on circles helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!