Facts that Matter: Circles | Mathematics (Maths) Class 10 PDF Download

Tangent to a Circle

A tangent to a circle is a line that touches the circle at only one point.

NOTE:

• There is only one tangent at each point of the circle.
• A tangent is a special type of secant, where the two ends of its chord are the same point.
• A tangent touches the circle at one point, while a secant cuts through at two points.
• The line that goes through the radius at the contact point is sometimes called the normal to the circle at that point.
• Therefore, at any point on the circle, there can only be one tangent.

Theorem 1

The tangent at any point of a circle is perpendicular to the radius, through the point of contact.
Proof: We have the centre O of the given circle and XY is the tangent to the circle at a point P.
Let us take a point Q on XY other than P. Join OQ.
Obviously, Q lies outside the circle. i.e., OQ > OP

Since, all the points on XY, except P lies outside the circle.
i.e., OP is smaller than all the distance of the point O from XY.
i.e., OP is the smallest distance of O from XY.
i.e., OP ⊥ XY