Circles Class 10th Notes - Free NCERT Class 10 Maths Chapter 10 Notes - Download PDF

A circle is a shape or geometrical shape that consists of all points in a plane that are equidistant from a point called the center. It is a round shape closed figure that does not have any edges or sides, or corners. It is a set of points, and these points have the same distance from a point. There are many examples of circles in the natural world, like the moon, the sun, the pupil of the eye and as well as human-made objects like plates, lenses, CDs, DVDs, etc. Circles are also used in architectural design and engineering, like, designing circular objects like gears and other mechanical components.

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These notes cover basic definitions of circle, its topics, subtopics and related formulae and theorems. These notes include types of lines in a circle, tangent, secant, parallel tangent, and determining the length of different types of tangents. CBSE Class 10 chapter on circles includes all the topics covered in these notes. Students must practice questions and check their solutions using the NCERT Solutions for Class 10 Maths Chapter 10 Circles. The NCERT class 10th maths notes are designed by our experts, which contain definitions, formulas, and examples for each chapter. These NCERT notes can be downloaded according to the standard and subjects.

NCERT Class 10 Maths Chapter 10 Circles: Notes

Circle: A circle is a 2-dimensional closed geometrical shape that contains a set of points, and all points are at a fixed or constant distance from a point called the center. All the points of the surface of the circle are at an equidistance.

Radius: The distance from the center of the circle to any point on the surface of the circle is called the radius of the circle. It is denoted by r.

Diameter: Any straight line segment that passes through the center of the circle and whose endpoints lie on the surface of the circle is called a diameter. It is denoted by d.

Circle and Line in a Plane:

There are three possible places for a line segment and the circle:
1. Non-intersecting: In this case, there is no common point between the line segment and the circle. As shown in Figure (i)

2. Touching: In this case, the line segment touches the surface of the circle at only one point, as shown in figure (ii).

3. Intersecting: In this case, the line segment intersects the surface of the circle at two points, as shown in figure (iii).

Tangent:

When any straight line touches the surface of the circle at exactly one point, it is called a tangent to the circle. As shown in the figure, AB is a tangent that touches the surface of the circle at only one point, C.

Secant:

When a straight line intersects the surface of the circle or has two common points with the surface of the circle, then it is called a secant of the circle. As shown in the figure, AB is a secant that intersects the surface of the circle at points P and Q.

A Special Case of Secant:

A special case of the secant is called the tangent of the circle when the two endpoints of its corresponding chord coincide. As shown in the figure, CC' is the special case of the secant.

Two Parallel Tangents at Most for a Given Secant

If we try to draw some parallel tangents to the surface of the circle for every given secant, then there will be two tangents that touch the circle at two diametrically opposite points and are parallel to it. As shown in the figure, AB is a secant of the circle and A'B' and A''B'' are parallel tangents.

Theorems: Tangent Perpendicular to the Radius at the Point of Contact

Theorem: The tangent to the circle at any point is the perpendicular to the circle's radius through the point of contact.

Here, OP is perpendicular to AB.

Assume a circle with centre “O” and AB being the tangent to the circle at point “P”. Now, according to the theorem, we have to prove that OP is perpendicular to the tangent AB.
Make a point Q outside the circle but on the tangent line AB, at the other place of P and join O and Q as shown in the figure. Otherwise, if Q lies inside the circle, then AB will become a secant of a circle.
As shown in the figure, Q lies outside the figure.
Therefore, OQ > OP
This is for every point that lies outside the circle, except point P, which lies on the line AB. It implies that OP is the shortest of all the points from point O to the tangent line AB.
Therefore, OP is perpendicular to AB.

Drawn The Number of Tangents from a Given Point:

There are three possible cases to draw tangents from a given point on the circle:
1. If a point lies inside or interior region of the circle, then any line that passes through the point will be a secant. It implies that no tangent that passes through the point in the interior region of the circle is possible.

2. When there is a point lying on the surface of the circle, then only one possible tangent can be drawn that passes through that point.

3. If a point lies outside the circle, then there are exactly two tangents that can be drawn that pass through that point, as shown in the figure.

Length of a Tangent:

If an external point is given, A is given, and a point B is given at the surface of the circle, then the line segment AB is called the length of the tangent from point A to the circle.

The length of the tangent using the Pythagorean theorem:
l² = d² - r²
Where,
I = The length of the circle
d = The external point from the center of the circle
r = radius

Lengths of Tangents Drawn from an External Point:

Theorem: The lengths of tangents drawn from an external point to a circle are equal.

In the figure, a circle is given with center O, and a point A is an exterior point of the circle with two tangents AB, AC on the circle from A.
According to the theorem, we need to prove AB = AC
So, we need to join OA, OB and OC.
We know that the angles between the radii and tangents form a right-angled triangle.
Therefore,
∠OBA and ∠ OCA are right angles
In right triangles OBA and OCA,
OB = OC (Radii of the same circle)
OA = OA (Common in both triangles)
Therefore, ∆ OBA ≅ ∆ OCA (RHS)
This gives PQ = PR

Important Properties Of a Circle:

1. If two concentric circles are given that the chord of the larger circle that touches the smaller circle is bisected at that contact point.
2. At the end of the diameter, when tangents are drawn then these tangents are parallel to each other.
3. The shape that is drawn circumscribing a circle is a rhombus

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