**Tangents to Circles Worksheet :**

Worksheet given in this section is much useful to the students who would like to practice problems on tangents to circles.

Before look at the worksheet, if you would like to know more about the stuff tangents to circles,

**Problem 1 : **

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

**Problem 2 : **

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

**Problem 3 :**

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

**Problem 4 : **

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

**Problem 5 : **

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

**Problem 6 :**

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 ?

**Problem 7 : **

In the diagram shown below,

SR is tangent at R to the circle with center P

ST is tangent at T to the circle with center P

Prove : SR ≅ ST

**Problem 8 :**

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.

**Problem 1 : **

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.

**Problem 2 : **

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.

**Problem 3 :**

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.

**Problem 4 : **

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.

**Problem 5 : **

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, EF is tangent to the circle with center at D.

**Problem 6 :**

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, 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.

**Problem 7 : **

In the diagram shown below,

SR is tangent at R to the circle with center P

ST is tangent at T to the circle with center P

Prove : SR ≅ ST

**Solution : **

**Problem 8 :**

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, 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 Worksheet".

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