Tangent of a Circle

Introduction

Tangent to a circle is line that touches circle at one point. A collection of points that are equidistant from a fixed point is called a circle whereas a straight group of points in a plane is called a line. In geometry, if a line in the same plane as that of the circle, touches the circumference of the circle outside the circle at exactly one point, that is, there is only one point of intersection between the two, then that line is called a tangent of the circle. Infinite number of such lines can be drawn from a point outside of a circle.

Tangent of a circle

A line touching a circle at exactly one point in its circumference in the same plane is called a tangent of the circle. The single common point between the line and the circle is called the point of tangency.

This is a geometrical representation of a circle with centre O intersecting a line AC at a point B in the plane. Since the line and the circumference of the circle share a single point in common, therefore, AC becomes the tangent of the circle with B as the point of tangency. This line lies outside the boundary of the circle.

The tangent to a circle with the equation x²+y²=a² at (p,q) is given by p²+q²=a².

Properties of Tangent of a circle

There are various properties that define the existence of tangents to a circle at a particular point. Some of these properties are discussed below −

PROPERTIES	GEOMETRICAL REPRESENTATION
A tangent to a circle touches the circle on its circumference at exactly one point.
There are only two tangents parallel to the secant of a circle.
Two tangent lines can be drawn to a circle from an external point.

Theorems Related to Tangent of a Circle

1. Tangent at any point of a circle always makes a 90° angle with the radius of the circle.

Proof : Take a point M on the tangent AB of the circle with centre O. This point M lies outside the circle because a tangent and a circle share only one point in common and that is P. Therefore, OM is longer than OP (radius of the circle).

$$\mathrm{OM > OP}$$

This is true for all points except P on AB.

⇒ OP is the shortest distance from point O to any point on AB.

Since the perpendicular distance is the shortest distance between a point and a line, OP is perpendicular to AB.

2.Two tangents drawn from an external point to a circle are of equal length.

To Prove : AB=AC

Proof : A is a point lying outside the circle with centre O. AB and AC are two tangents from A to the circle at B and C, respectively.

Since we know that tangent to a circle is perpendicular to the radius,

$$\mathrm{∴ ∠OBA=∠OCA}$$

⇒ OBA and OCA are right angle triangles.

Now,

In OBA and OBC,

OB=OC (Radius of the circle)

OA=OA (Common)

$$\mathrm{\angle OBA=\angle OCA=90°}$$

$$\mathrm{∴ △OBA≅△OCA}$$

$$\mathrm{⇒ AB=AC}$$

Hence, two tangents drawn from an external point to a circle are of equal length.

Solved Examples

1. Find the length of the tangent PB in cm if the length of the radius of the circle is 3cm and OB = 5 cm.

Since OPB is a right angle triangle because tangent to a circle is perpendicular to the radius at the point of tangency,

$$\mathrm{∴ (OP)^2+(PB)^2=(OB)^2}$$

$$\mathrm{\Rightarrow (3)^2+(PB)^2=(5)^2}$$

$$\mathrm{\Rightarrow (PB)^2=(5)^2-(3)^2}$$

$$\mathrm{\Rightarrow (PB)^2=16}$$

$$\mathrm{\Rightarrow PB=4cm}$$

2.In the given figure, a circle with centre O touches the sides of $\mathrm{\angle ACB}$ at A and B. If$\mathrm{\angle BAC=80°}$, find the measure of $\mathrm{\angle BOC}$

$$\mathrm{\angle BAC+\angle ACO+\angle COB+\angle OBA=360°}$$

This is because the sum of angles in a quadrilateral is equal to 360°.

$$\mathrm{80°+90°+90°+ \angle BOC= 360°}$$

$$\mathrm{\angle BOC=100°}$$

3. In the given figure,$\mathrm{\angle BOC=140°}$, Find $\mathrm{\angle BAC}$

Since the size of an inscribed angle is half the size of the central angle that forms the same arc,

Therefore,

$$\mathrm{\angle BEC=1/2 \angle BOC}$$

$$\mathrm{\Rightarrow \angle BEC=70°}$$

Now,

$$\mathrm{\angle BAC+\angle ACO+\angle COB+\angle OBA=360°}$$

$$\mathrm{\Rightarrow \angle BAC=360°-90°-90°-70°}$$

$$\mathrm{\Rightarrow \angle BAC=110°}$$

Conclusion

The tutorial describes the tangent of a circle as the single most common point between a line intersecting a circle in the same plane. A circle can have infinitely many tangents, however, only one tangent can be drawn to a circle from a single point. The tangent is perpendicular to the radius of the circle, that is, it makes a 90° with the radius. The point at which the tangent touches the circle is called the point of tangency. The tangent to a circle is a valuable information as it enables us to find out the slope of a curved function at a specific point at the curve. The tangent to a circle with the equation x²+y²=a² at (p,q) is given by p²+q²=a².

FAQs

1. Can two circles be tangent to each other?

A tangent is a line touching a circle at exactly one point. So, if a circle touches another circle in the same plane at exactly one point, then they share one point in common, hence, one circle is tangent to another at that point.

2. What is the difference in the number of tangents that can be drawn from a point on the circumference of the circle and from an external point?

From a point on the circumference of the circle, only one tangent can be drawn because a tangent touches the circle at exactly one point. However, from an external point,two tangents can be drawn to the circle that touches the circle at the point each.

3. Give two real life examples of the tangent of a circle.

• A cycle tire is a circle moving on a plane road where the road forms a tangent at each point on the tire as the cycle moves.

• A ball rolling on a see-saw is another example where the surface of the see-saw is tangent at each point on the ball as it slides down the see-saw.

4. What is the smallest distance between the centre of a circle and its tangent?

The smallest distance between the centre of a circle and its tangent is the radius of the circle because any other point on the tangent of the circle lies outside the boundary of the circle which means it is more than the radius of the circle.

5. Can a tangent be a secant?

A tangent to a circle can be a special case of secant where the two end points of its respective chord coincide with each other.