Circle

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.

(Image will be Uploaded Soon)

Circle 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.

(Image will be Uploaded Soon)

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

Arc Length of a Curve

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.

(Image will be Uploaded Soon)

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.

(Image will be Uploaded Soon)

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.

(Image will be Uploaded Soon)

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.

(Image will be Uploaded Soon)

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.

(Image will be Uploaded Soon)

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

(Image will be Uploaded Soon)

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.

(Image will be Uploaded Soon)

Properties of Tangent of a circle

In geometry, the tangent is defined as the line or plane that just touches a curve or a curved surface at exactly 1 point. The point of tangency is defined as the only 1 point of intersection where the straight line touches. the circle. Some properties of Tangent are- 

•  The tangent of a circle touches the circumference at only 1 point.

• When the tangent touches the circle’s radius, it makes a right angle with it.

• The line of Tangent of a circle never enters the interior of that circle.

Arc Length

The arc length is the length of the part of the circumference that an arc covers. This can be calculated using different methods and formulas based on the data we have. The ways to solve it are-

• When the Radius and Central Angle Is Given -

The arc length of a circle, when the radius and central angle is given, can be found out using the following arc length formula,

Length of a given Arc = θ × r (where θ is in radian)

Length of a given Arc = θ × (π/180) × r (where θ is in degree)

• When the Radius Is Not Given -

The arc length of a circle when the radius is not given can be found out in 2 ways. These are-

Using Central Angle and the Sector Area:

Start with first multiplying the sector area by 2 and dividing that result by the central angle (in radians). Then the next step is to find the square root of the result of the previous division and get the product of the obtained root, with the central angle again to get the arc length. The units of the arc length will be the square root of the sector area units.

Using Central Angle and the Chord Length:

Start with dividing the central angle (in radians) by 2 and apply the sine function on it. Then divide the given chord length by two times the result of our step 1. This calculation gives the radius as result and multiplying it by the central angle will give the arc length.

• When the Central Angle Is Not Given-

The arc length of a circle, when the central angle is not given, can be found out in 2 ways. They are-

Using Radius and the Sector Area:

First, find the product of the sector area by 2 and then divide the result of it by the radius squared, after bringing the units to a common unit, to get the central angle( in radians). Then by multiplying the central angle by the radius will give the length of the arc.

Using Radius and Chord Length:

Start with dividing the chord length by twice the given radius and then find the inverse sine of the result of the above calculations. Then multiply the result of the inverse sine with 2. This will give the central angle (in radians.) Finally, multiply the above central angle by the radius to get the length of the arc.

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.”

(Image will be Uploaded Soon)

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.

(Image will be Uploaded Soon)

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 Is 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