Circle

Circle Introduction Class 10th

A circle is a two-dimensional figure formed by a set of all those points which are equidistant from a fixed point in the same plane. The fixed point is termed the center of a circle and a fixed distance from the center of the circle is termed as the radius of a circle. To help you to learn more about circles, we at Vedantu are providing Class 10th Circle Math notes here. In these Class 10th circle notes, you can find circle introduction Class 10th, circle theorem Class 10th, properties of circles Class 10th etc. These Class 10th circle Maths notes will be highly beneficial for the students preparing for the Class 10th Maths exam.

 

Circle for Class 10th

A circle is a two-dimensional figure which is measured with respect to its radius.. The circle splits the plane into two zones i.e. external zone and interior zone. The center point of the circle is the center of a circle from which all the distances to the points are equal.The circle has both area and perimeter. The perimeter of a circle is also considered as a circumference of a circle whereas the area of the circle is the region circumscribed by it in a 2D shape.

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Circle for Class 10th Definition

A circle is a closed shape two-dimensional in which the set of all points in the plane is equally spaced from the centre of the circle. Each line that moves through the circle forms the line of reflection symmetry. The circles also includes rotational symmetry around the center for every angle.The formula of the circle in a plane is derived as

 

(x-h) 2 + (y-k) 2 =r2

Here x and y are the coordinate points of a circle whereas h and k are the center of a circle.

 

Circle and Line Drawn on a Plane

There are Three Possibilities For a Circle and a Line Drawn on a Line. These Three Possibilities are:

  • Lines can be non-intersecting

  • The lines can have a single common  point. In this particular case, the line touches the circle

  • Lines can have two common points. In this particular case, the line is cutting the circle.

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In the above figure,

The Line is non- intersecting in the first image

The Line is touching in the  second image

The Line is intersecting in the third image

 

Tangent to a Circle

A tangent to a circle is a line segment that touches the circle at exactly one point. There is a unique tangent line which passes through every point on the circle.

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Secant

A secant to a circle is a line that has two common points in the circle. This line forms a chord of the circles by breaking the circle at two points.

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Tangent as an Exceptional Case of Secant

The tangent to a circle can be seen as an exceptional case of secant when the endmost points of its equivalent chord coincide.

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Two Parallel Tangents at Most for a Given Secant

For each given secant of a circle, there are exactly two tangents that are parallel to it and touch the circle at two diametrically opposite points.

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Number of Tangents Can be Drawn from a Given Point

Any line that passes through the point which lies in an interior region of a circle will be the secant of a circle. No tangent line can be drawn to the circle which passes through a point that lies in the interior region of a circle.

There will be exactly one tangent line that can be drawn to a circle passing through the point of tangency lying on the circle.

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There will be exactly two tangent lines that can be drawn to a circle passing through the point of tangency lying outside the circle.

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Length of a Tangent

The length of the tangent from any point (let's consider P) to the circle is defined as the measurement of the tangent from the point P to the point of tangency I with the circle. Here, PI is considered as the tangent length.

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Properties of a Circle Class 10th

Here are some of the properties of a circle class 10th 

  • The diameter of a circle is considered to be the longest chord of the circle

  • A Circle having a discrete radius are equal

  • A circle can be drawn in a rectangle, trapezium, square, kite, triangle etc.

  • The chord of a circle that is at equal distance from the center are equal in length

  • If the tangents of a circle are made at the end of the diameter, they are parallel to each other.

  • The radius of a circle drawn from a perpendicular to the chord bisects the chord.

  • Similar chords and similar circles have  similar circumference.

  • The circles are considered as a congruent if they have equal radii.

 

Solved Examples Based on Circle Theorem Class 10th

Here, are some of the solved examples on Circle theorem Class 10th;

1. Prove the statement “The tangent drawn to any point of a circle is perpendicular to the radius through the point of contact.”

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Given -A circle is drawn with centre O along with the tangent  ℓ  at a point of contact A

To prove - OA⊥ ℓ

 

Proof : Let B be any point on the line segment  ℓ

Construction- Join OB

Suppose the line touches at point C in the circle.

Hence,

OB > OC

OB > OA (as radius - OA =OC)

There will be similar case with all the other points on the circle

Hence, OA is the smallest perpendicular line that intersects ℓ.

∴ OA ⊥  ℓ

 

2. Prove the statement “ The length of tangent drawn from the exterior point of the circle is equal.

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Given: Let a circle be drawn with center O and T be any point on the exterior of the circle.

TP and TQ are two tangent to a circle intersecting at two points i.e. P and Q respectively.

 

To prove: Length of the given tangents are equal such that TP=TQ

Construction- Join OP, OQ ,and OT

 

Proof: As TP is tangent to a circle

OP⊥ TO

∠POT= 90°

Hence, ▲ POT is a right angle.

 

Fun Facts

  • The first theorem related to the circle was founded by Thales around 850 BC.

  • The circles do not include any angles

  • The word encyclopedia means “ the circle of learning”

  • The discovery of wheels which are circular in shape is one the most important history of humans.

 

Quiz Time

1. The Length of a Tangent Which are Drawn From an Exterior Point to the Circle.

  1. Are equal

  2. Not equal

  3. Occasionally equal

  4. Not defined.

 

2. Tangents Which are  Drawn From an Exterior Point  to a Circle are

  1. Always equal 

  2. Not equal

  3. Parallel 

  4. Perpendicular

 

3. Calculate the Length of a Tangent Drawn to a Circle with the Radius 7 cm From a Point  25 cm Away From the Center Point of the Circle.

  1. 24 cm

  2. 25 cm

  3. 26 cm

  4. 27 cm

FAQ (Frequently Asked Questions)

1. Explain the Basic Terminologies of a Circle.

Ans: A circle is a two-dimensional figure which is measured with respect to its radius. The circle is a figure which divides the plane into region i.e.external region and internal region. The perimeter of a circle is also known as the circumference of the circle whereas the area of the circle is the area bounded by it in a 2D plane.

Annulus The area bounded by two concentric circles is known as the  annulus of the circle. It is basically considered as a ring-shaped object.

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Arc- Arc is the connected curve of a circle.

Sector- The area bounded by two radii and arc is the sector of the circle.

Segment- The segment of the circle is the area bounded by a chord and arc lying between the endpoints of the chord. It is to be noted that the segment does not include any centre point.

Chord- Chord is a line segment that joins any two points on the circumference of the circle. The Diameter is the longest chord of a circle which passes through the center point of a circle.

Centre- The center is the midpoint of the circle

Radius- A radius is a straight line made from the center point of a circle to the circumference of a circle.

Diameter The line which passes through the centre of a circle having two extreme points on the circumference of a circle is known as the diameter of a circle. It makes the two semicircles by dividing the circle into two equal parts.

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2. What is a Secant of a Circle?

Ans: A secant of a circle is a straight line that intersects a circle in two points. The word secant came from the Latin word “secare”. A chord is a line segment that joins any two points on the circumference of the circle. A secant of a circle is an extension of a chord which is a straight line segment of which the ending points are located on the circle. If a similar chord passes through the center of a circle, then it becomes the diameter of a circle. So, an extended diameter is considered as a Secant.

Difference between the Secant of a Circle and the Chord

Secant

Chord

Secant intersects the circle at two points

Whereas chord touches the circle at two points

It is drawn from outside the circle

It lies within the circle


The Secant of a Circle Formula

If both secant and tangent of a circle drawn from  the point outside the circle then its formula will be derived as :

Length of the secant * its external segment = (length of the tangent segment)2