INTERNAL AND EXTERNAL TANGENTS OF A CIRCLE

Tell whether the common tangent(s) are internal or external.

Question 1 :

Solution :

When we draw segments from the center of the circle, it does not intersect the tangents. So, they are external tangents.

Question 2 :

Solution :

When we draw segments from the center of the circle, it intersects the tangents. So, it is internal tangent.

Question 3 :

Solution :

When we draw segments from the center of the circle, it intersects the tangents. So, they are internal tangents.

Copy the diagram. Tell how many common tangents the circles have. Then sketch the tangents.

Question 1 :

Solution :

We can draw 2 internal tangents and 2 external tangents.

Question 2 :

Solution :

We cannot draw any common tangents for the given circles.

Question 3 :

Solution :

We can two common tangents for the circles given above and they are external tangents.

Are these lines internally or externally tangent? Connect the centers and apply the definitions. Are the circles tangent internally or externally?

Question 4 :

Solution :

(i)  The line segment joining the centers is intersecting the the common tangent. So, it is internal tangent.

(ii)  The line segment joining the centers is not intersecting the the common tangent. So, it is external tangent.

Question 5 :

In the diagram P and Q are tangent circles. RS is a common tangent. Find RS.

Solution :

RS is a external tangent.

Tangents are perpendicular to the radius  at the point of tangency.

<QRS  =  <PSR  =  90 degree

We can connect a point on RQ from P. So, it will create the rectangle PSRT.

In triangle TPQ,

QP²  =  TQ² + TP²

8²  =  2² + TP²

64-4  =  TP²

TP  =  √60

TP  =  2√15

RS  =  2√15

Question 6 :

If two circles are internally tangent, what is the total number of common tangents that can be drawn to the circle ?

a)  1      b)   2     c)  3      d)  0

Solution :

In the picture shown above, we see two circles touches each other internally. So, maximum number of tangents can be drawn is 1.

Question 7 :

JH is tangent to a circle G at J. Find the value of x.

Solution :

By observing the circle above, GK and GJ are radii of the circle.

JH is the tangent and the line drawn from the center of the circle to the point of tangency, then it must be perpendicular.

GH² = GJ² + JH²

(x + 8)² = x² + 12²

x² + 2x(8) + 8² = x² + 144

16x + 64 = 144

16x = 144 - 64

16x = 80

x = 80/16

x = 5

So, the radius of the circle is 5 cm.

Question 8 :

AB and AC are tangents to circle O. AB = 4x² - 42, AC = 17x and OC = 6x + 15. Find OC.

Solution :

Given that,  AB = 4x² - 42, AC = 17x

In general, the length of tangents drawn from the external point of the circle will be equal.

AB = AC

4x² - 42 = 17x

4x² - 17x - 42 = 0

4x² - 24x + 7x - 42 = 0

4x(x - 6) + 7(x - 6) = 0

(4x + 7)(x - 6) = 0

Equating each factor to 0, we get

To find the length of OC, we have to apply the value of x in the expression OC. Since one of the value is negative, we cannot accept that.

OC = 6(6) + 15

OC = 36 + 15

= 51

So, the radius is 51 cm.

Question 9 :

Two circles have a common external tangent. The radius of one circle is 10 cm. The radius of the other circle is 19 cm. The distance between the centers of the circles is 41 cm. Find the length of the common tangent.

Solution :

By understanding the information, we draw the picture given below.

In triangle ACQ,

CQ = x, PC = 9 cm and PQ = 41

It is a right triangle. In every right triangle Pythagorean theorem will exists.

PQ² = PC² + CQ²

41² = 9² + x²

x² = 1681 - 81

x² = 1600

x = √1600

x = 40

So, the length of CQ is 40 cm.

Question 10 :

AB and AC are tangents of circle with center O and OC = 5x. Find OC.

Solution :

Tangents drawn from the external point will have equal length.

AB = AC

14 + 4x = 19 - 6x

4x + 6x = 19 - 14

10x = 5

x = 5/10

x = 1/2

OC = 5x

= 5(1/2)

OC = 5/2 ==> 2.5

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