Relationship Between Radius of Circle And Tangent


 
 
Concept Explanation
 

Relationship Between Radius of Circle And Tangent

Relationship between Radius of circle with tangent:

Theorem:   The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Given : A circle with center O. XY is a tangent and P is the point of contact.

To Prove: OP is perpendicular to XY

Construction: Take a point Q on XY join OQ.

Proof: Now point Q lies outside the circle because the tangent just touches the circle at point P

So as Q points lies on XY therefore it is outside the circle.

  OQ > radius of the circle  [ Q is outside the circle]

  OQ > OP

So any point on XY the distance from the center will be more than radius

That is OP is the shortest of all the distances of the point O to the points on XY.

So, OP is perpendicular to XY   [ because the shortest distance is perpendicular] 

I

Illustration:   A point P is 13 cm from the centre of the circle. The length of the tangent from P to the circle is 12 cm. Find the radius of the circle.

Solution    Since the tangent to a circle is perpendicular to the radius through the point of contact,

                                large angle OTP=90^{circ}

 It is given that OP = 13 cm,   PT = 12 cm and let  OT =  r cm

In right triangle OTP,

  large OP^{2}=OT^{2}+PT^{2};;;Rightarrow ;;13^{2}=12^{2}+r^{2}

large Rightarrow ;169=144+r^{2};;;Rightarrow ;;r^{2}=25

large therefore   radius of the circle is 5 cm.

Sample Questions
(More Questions for each concept available in Login)
Question : 1

The length of the tangent drawn from a point 8 cm away from the centre of a circle, of radius 6 cm, is _______________

Right Option : C
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Question : 2

From a point A, the length of a tangent to a circle is 8cm and distance of A from the circle is 10cm. The length of the diameter of the circle is

Right Option : A
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Explanation
Question : 3

PT is a tangent to a circle whose centre is O. If PT = 12 cm and PO = 13 cm then find the radius of the circle.

Right Option : A
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Explanation
 
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