**Tangents to Circles :**

A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Line k in the diagram above is a tangent.

**Point of Tangency :**

The point where a tangent line touches the circle. Point m in the diagram above is the point of tangency.

**Common Tangent : **

A line or segment that is tangent to two coplanar circles is called a common tangent.

**Theorem 1 :**

If a line is tangent to circle, then it is perpendicular to the radius drawn to the point of tangency.

In the diagram shown below, if l is tangent to circle Q at P, then l ⊥ QP.

**Theorem 2 :**

If a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

In the diagram shown below, if l ⊥ QP at P, then l is tangent to circle Q.

From a point in a circle's exterior, we can draw exactly two different tangents to the circle. The following theorem tells us the segments joining the external point to the two points of tangency are congruent.

**Theorem 3 : **

If two segments from the same exterior points are tangent to a circle, then they are congruent.

In the diagram shown below, if SR and ST are tangent to circle P, then SR ≅ ST.

**Example : **

Tell which line or segment is best described as a tangent in the diagram shown below.

**Solution : **

EG is a tangent, because it intersects the circle in one point.

**Example 1 : **

In the diagram shown below, tell whether the common tangents are internal or external.

**Solution : **

The lines j and k intersect CD, they are common internal tangents.

**Example 2 : **

In the diagram shown below, tell whether the common tangents are internal or external.

**Solution : **

The lines m and n do not intersect AB, so they are common external tangents.

**Example : **

In the diagram shown below, describe all common tangent and identify the point of tangency.

**Solution : **

The vertical line x = 8 is the only common tangent of the two circles.

The point of tangency is (8, 4).

**Note : **

The point at which a tangent line intersects the circle to which it is tangent is the point of tangency.

**Example : **

In the diagram shown below, say whether EF is tangent to the circle with center at D.

**Solution : **

We can use the Converse of the Pythagorean Theorem to say whether EF is tangent to circle with center at D.

Because 112 + 602 = 612, ΔDEF is a right triangle and DE is perpendicular to EF.

So by Theorem 2 given above, EF is tangent to the circle with center at D.

**Example :**

In the diagram shown below, I am standing at C, 8 feet from a grain silo. The distance from me to a point of tangency is 16 feet. What is the radius of the silo ?

**Solution : **

By the Theorem 1 given above, tangent BC is perpendicular to radius AB at B. So ΔABC is a right triangle. So we can use the Pythagorean theorem.

Pythagorean Theorem :

(r + 8)^{2} = r^{2} + 16^{2}

Square of binomial :

r^{2} + 16r + 64 = r^{2} + 256

Subtract r^{2} from each side :

16r + 64 = 256

Subtract 64 from each side :

16r = 192

Divide each side by 16.

r = 12

Hence, the radius of the silo is 12 feet.

**Example : **

In the diagram shown below,

AB is tangent at B to the circle with center at C

AD is tangent at D to the circle with center at C

Find the value of x.

**Solution : **

By the Theorem 3 given above, two tangent segments from the same point are congruent.

AB = AD

Substitute :

x^{2} + 2 = 11

Subtract 2 from each side.

x^{2} = 9

Take square root on each side.

x = ± 3

Hence, the value of x is 3 or -3.

After having gone through the stuff given above, we hope that the students would have understood, "Tangents to Circles".

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