Haryana State Board HBSE 10th Class Maths Notes Chapter 10 Circles Notes.

## Haryana Board 10th Class Maths Notes Chapter 10 Circles

**Introduction**

We have studied in class IX that a circle is a collection of all the points in a plane which are at a constant distance from a fixed point. The constant distance is called the radius and fixed point is known as the centre. We have also studied various terms related to a circle like chord, sector, segment, arc etc.

In this chapter we shall learn about tangents to a circle and some of their properties. Let us now examine the different situations that can arise when a line and a circle are given in a plane.

**Tangent of a circle**

Consider a circle with centre O and a line P in the plane of the circle. There are three types of possibilities arise as shown in the figures below:

In the figure (i), the line P and the circle have no common point hence, the line P is known as a non intersecting line with respect to the circle.

In the figure (ii), the line P intersects the circle in two distinct points A and B. It is called a secant of the circle.

In the figure (iii), the line P intersects the circle in one and only one point A and is said to be a tangent to the circle. The point A at which the tangent line meets the circle is called the Point of contact.

Normal: The line containing the radius through the point of contact is also sometimes called the normal to the circle at the point.

Supplementary angles: Two angles having sum 180° are called supplementary angles.

Co-interior angles: Interior angles on the same side of the transversal are called co-interior angles or consecutive interior angles.

Concentric circles: Circles having the same centre are called concentric circles.

Parallelogram: A quadrilateral having each pair of opposite sides equal and parallel is called parallelogram.

Rhombus : A parallelogram having all the sides equal is called a rhombus.

Circumscribed circle: The circumscribed circle or circumcircle of a polygon is a circle passing through all the vertices of the polygon. The centre of this circle is called circumcentre and its radius is called the circumradius.

Inscribed circle: Inscribed circle or incircle of a polygon is the largest circle that can be contained in the polygon and it touches each side of the polygon at a point. Hence each side of the polygon is a tangent to the incircle. The centre of this circle is called incentre and its radius is called inradius.

A line which intersects the circle at only one point is known as the tangent to the circle. In the given figure PQR is a tangent to the circle and point Q is the point of contact.

The word tangent to a circle has been derived from the latin word “Tangere”, which means ‘to touch’ and was introduced by the Denish mathematician Thomas Fineke in 1583.

The tangent to a circle is a special case of the secant, when the two end points of its chord coincide.

**Some Properties of tangent to a circle**

Theorem 10.1:

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Given: A circle with centre O and a tangent AB to the circle at a point P.

To Prove: OP ⊥ AB.

Construction: Take any point R, other than P on the tangent AB. Join OR. Suppose OR meets the circle at Q.

Proof :

OP = OQ (Radii of the same circle)

But OP < OQ + QR

OP < OR

Thus, OP is shorter than any other line segment joining O to any point of AB, other than P.

We know that among all line segment joining the point O to a point on AB, the shortest one is perpendicular to AB.

Hence, OP ⊥ AB. Proved

**Number of Tangents from a point on a circle**

To get an idea of the number of tangents from a joint on a circle, let us perform the following activity:

From figure (III) PR_{1} and PR_{2} are two tangents drawn from a point P lying outside the circle. These tangents touch the circle at R_{1} and R_{2} respectively. So, R_{1} and R_{2} are called points of contact of tangenta PR_{1} and PR_{2}.

(i) There is no tangent to a circle passing through a point lying inside the circle.

(ii) There is one and only one tangent to a circle passing through a point lying on the circle.

(iii) There are exactly two tangents to a circle through a point lying outside the circle.

Length of Tangent: The length of segment of the tangent from the external point P and point of contact with the circle is called the length of the tangent from the point P to the circle.

From figure (III), PR_{1} and PR_{2} are the length of tangents from P to the circle.

Theorem 10.2:

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

Given: AP and AQ are two tangents from a point A to a circle C (O, r).

To Prove: AP = AQ

Construction: Join OA, OP and OQ.

Proof: AP is a tangent at P and OP is the radius through P.

Similarly AQ is a tangent at Q and OQ is the radius through Q.

∴ OQ ⊥ AQ

In the right ΔOPA and ΔOQA, we have

OP = OQ [equal radii of the same circle]

AO = AO [common]

∠OPA = ∠OQA [each is 90°]

∴ ΔOPA ≅ ΔOQA [By RHS congruence]

⇒ AP = AQ [By CPCT Proved]

Hence, AP = AQ. Proved