Construct A Tangent To A Circle

Tangent to a circle

Tangent of a circle is a line which touches the circle exactly at one point. The point at which tangent touches the circle is known as ‘point of contact’. Radius of the circle and tangent are perpendicular to each other at the point of contact.

Tangent To A Circle

We know that, there cannot be any tangent drawn to the circle through a point inside the circle.

There can be only one tangent to a point on the circle.

Let’s see how to draw a tangent to a circle at a point on the circle. (Refer fig.)

  • Draw a circle with required radius with center \(O\)
  • Join center of the circle \(O\) and any point P on the circle. \(OP\) is radius of the circle.
  • Draw a line perpendicular to radius \(OP\) through point \(P\). This line will be a tangent to the circle at \(P\).

Tangent To A Circle

Two tangents can be drawn to a circle from a point outside of the circle. Lengths of the two tangents will be equal.

How to draw two tangents from a point outside of the circle is discussed below.

Consider a point A outside the circle with center \(O\).

  • Join the points A and O, bisect the line \(AO\). Let \(P\) be the midpoint of \(AO\).
  • Tangent To A CircleDraw a circle taking \(P\) as center and \(PO\) as radius. This circle will intersect at two points \(B\) and \(C\) on the circle with center \(O\).
  • Tangent To A CircleJoin point \(A\) with \(B\) and \(C\). \(AB\) and \(AC\) are the required tangents through points \(B\) and \(C\) on the circle.

Tangent To A Circle

Why \(AB\) and \(AC\) are tangents to the circle with center \(O\)?

Join \(BO\).

Tangent To A Circle

It is observed that \(AO\) is diameter of the circle with center \(P\). By construction, \(∠ABO\) is an angle in semi-circle.

Therefore,

\(∠ABO\) = \(90°\)

Since, \(OB\) is radius of the circle with center \(O\), \(AB\) has to the tangent through the point \(B\).

Similarly, \(AC\) is the tangent through the point \(C\).

  1. When the center of the circle is not given to draw tangent to the circle from a point outside the circle (Refer fig),
  • Draw two non-parallel chords \(AB\) and \(CD\).
  • Draw perpendicular bisector for both \(AB\) and \(CD\).
  • The point \(O\) at which these bisectors intersect will be the center of the circle because, radius is perpendicular bisector to chord.

Tangent To A Circle

  • Now, follow the steps to draw tangents when the center is given.

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Practise This Question

A circle of radius r has a centre O. What is first step to construct a tangent from a generic point P which is at a distance r from O?