Circle Theorems on Tangents

A line which is drawn from an external point of a circle and meets at one and only one point on the circumference of the circle is called a tangent. The point where the tangent touches the circle is called the point of contact.

In the given figure, TAN is a line which meets the circle at a point A. So, TAN is a tangent and point A is the point of contact.

 

In the same figure, a line TB cuts the circle at two points D and E. So, the line TB is called a secant line.

 

 

Properties of Tangent

 

(i)           The length of tangents drawn from an external point to a circle are equal.

(ii)        The tangent at any point of a circle is perpendicular to the radius through the point of contact.

(iii)     The angles in the alternate segments are always equal.

 

 

THEOREM 10:

“Tangents drawn from an external point on a circle are equal in length.”

 

This theorem can be verified by an experiment. Here is the experimental verification of the theorem:

 

Experimental Verification:

 

Construction: By using a pencil and compass, two circles of different radii with a centre at ‘O’ are drawn. In each figure, radii OA and tangent TAN are drawn.

 

With the help of a protractor, ∠OAT and ∠OAN of each figure are measured and the results are tabulated below:

 

Table:

  

Conclusion: The above experiment shows that the tangent drawn from an external point on a circle are equal in length.

 

 

THEOREM 11:

“Line segment joining centre of the circle and point of contact is perpendicular to the tangent.”

 

This theorem can be verified by an experiment. Here is the experimental verification of the theorem:

 

Experimental Verification:

 

Construction: By using a pencil and compass, two circles of different radii with a centre at ‘O’ are drawn. In each circle, from an external point P two tangents PAT₁ and PBT₂ are drawn.

With the help of a divider and a ruler, the length PA and PB of tangents from the external point P to the points of contact A and B of each figure are measured and the results are tabulated below:

 

Table:

Conclusion: The above experiment shows that the tangent is perpendicular to the radius of the circle at the point of contact.

 

 

THEOREM 12:

“The angles in the alternate segments are always equal.”

 

This theorem can be verified by an experiment. Here is the experimental verification of the theorem:

 

Experimental Verification:

 

Construction: By using a pencil and compass, two circles of different radii with a centre at ‘O’ are drawn. In each circle, tangent TAN and alternate segment angles ∠ABC and ∠CAN are drawn.

With the help of a protractor, the angles ∠ABC and ∠CAN are measured and the results are tabulated below:

 

Table:

  

Conclusion: The above experiment shows that the angles in the alternate segments are equal.