A line that joins two close points from a point on the circle is known as a tangent. In simple words, we can say that the lines that intersect the circle exactly in one single point are tangents. Only one tangent can be at a point to circle. The point where a tangent touches the circle is known as the point of tangency. The point where the circle and the line intersect is perpendicular to the radius. As it plays a vital role in the geometrical construction there are many theorems related to it which we will discuss further in this chapter.

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Here, point O is the radius, point P is the point of tangency.

Only when a line touches the curve at a single point it is considered a tangent. Or else it is considered only to be a line. Hence, we can define tangent based on the point of tangency and its position with respect to the circle.

When point lies on the circle

When point lies inside the circle

When point lies outside the circle

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Here, from the figure, it is stated that there is only one tangent to a circle through a point that lies on the circle.

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In the figure above, the point P is inside the circle. Now, all the lines passing through point P are intersecting the circle at two points. therefore, no tangent can be drawn to the circle that passes through a point lying inside the circle.

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The above figure concludes that from a point P that lies outside the circle, there are two tangents to a circle.

Always remember the below points about the properties of a tangent

A line of tangent never crosses the circle or enters it; it only touches the circle.

The point at which the lien and circle intersect is perpendicular to the radius

The tangent segment to a circle is equal from the same external point.

A tangent and a chord forms an angle, the angle is exactly similar to the tangent inscribed on the opposite side of the chord.

Below is the equation of tangent to a circle

Tangent to a circle equation x2+ y2=a2 at (a cos θ, a sin θ) is x cos θ+y sin θ= a

Tangent to a circle equation x2+ y2=a2 at (x1, y1) is xx1+yy1= a2

Tangent to a circle equation x2+ y2=a2 for a line y = mx +c is y = mx ± a √[1+ m2]

Tangent to a circle equation x2+ y2=a2 at (x1, y1) is xx1+yy1= a2

To understand the formula of the tangent look at the diagram given below.

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Here, we have a circle with P as its exterior point. From the exterior point P the circle has a tangent at Point Q and S. A straight line that cuts the curve in two or more parts is known as a secant. So, here the secant is PR and at point Q, R intersects the circle as shown in the diagram above. So, now we get the formula for tangent-secant

PR/PS = PS/ PQ

PS² = PQ.PR

Theorem 1

A radius is gained by joining the centre and the point of tangency. A tangent at the common point on the circle is at a right angle to the radius. The below diagram will explain the same where AB \[\perp\] OP

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Theorem 2

From one external point only two tangents are drawn to a circle that have equal tangent segments. A tangent segment is the line joining to the external point and the point of tangency. According to the below diagram AC = BC

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Example 1

In the below circle point O is the radius, PT is a tangent and OP is the radius, If PT is a tangent, then OP is perpendicular to PT.

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If OP = 3 Units and PT = 4 Units. Find the length of OT

Solution: as the radius is perpendicular to the tangent at the point of tangency, OP \[\perp\] PT

Therefore, ∠P is the right angle in the triangle OPT and triangle OPT is a right angle triangle.

Now, according to the Pythagoras theorem, we find OT.

(OP)² + (PT)² = (OT)²

3² + 4² = (OT)²

9 + 16 = (OT)²

25 = (OT)²

5 = OT

As the length cannot be negative, the length of OT is 5 units.

Example 2

In the below diagram PA and PB are tangents to the circle. Find the value of

∠OAP

∠AOB

∠OBA

∠ASB

The length of OP, PB = 7 cm (given)

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Solution:

∠OAP = 90° (Tangent is perpendicular to the radius)

∠AOB + ∠APB = 180°

∠AOB + 48° = 180°

∠AOB = 180° - 48° = 132°

∠OBA + ∠OAB + ∠AOB = 180° (angle sum of triangle)

2 x ∠OBA + ∠AOB = 180° (∠OBA = ∠OAB)

2 x ∠OBA + 132° = 180° (∠AOB = 132°)

∠OBA = 24°

∠AOB = 2 x ∠ASB (angle at centre = 2 angle at circle)

∠ASB = ∠AOB / 2

∠ASB = 132° / 2 = 66°

Cos 24° = \[\frac{7}{OP}\] ⇒ OP = \[\frac{7}{cos24^{0}}\]

FAQ (Frequently Asked Questions)

1. How to Know if Two Circles are Tangent?

When a line is tangent to a circle it indicates that the line is touching the circle at a single point. If a circle is tangent to another circle, it shows that the two circles are touching each other at exactly the same point. There is an interesting property when two circles are tangent to each other. The common tangent line will be perpendicular to both the radii of the two circles at a common point. This means that the three points (the 2 radii and the tangent point) will lie on a straight line. If you draw a line connecting these three points, you will end up with a straight line.

2. What do you Mean When you say the Lines are Tangent?

If any line touches a curve at a point and does not crossover or penetrate the circle, or touches it at any other point, then, it is a tangent line. Such a line also displays another characteristic. It is a line that crosses a differentiable curve at a point where the slope of the curve equals the slope of the line. Moreover, a line that is tangent to a circle forms a perpendicular at the radius to the point of tangency. This happens irrespective of which point of the circle touches the tangent line.