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Note :

If centre of the circle is not given, (Ex 11.2, 7)

We find its center first by

  1. Taking any two non-parallel chords
  2. And then finding the point of intersection of their perpendicular bisectors.

Then you could proceed as Construction 11.3 .

-v-

Construction 11.3:

To construct the tangents to a circle from a point outside it.

We are given a circle with centre O and a point P outside it. We have to construct the two tangents from P to the circle.

Steps of Construction :

  1. Join PO and bisect it. Let M be the midpoint of PO.
  2. Taking M as centre and MO as radius, draw a circle.
  3. Let it  intersect the given circle at the points Q and R.
  4. Join PQ and PR.

Then, PQ and PR are the required two tangents.

Justification:

We need to prove that PQ and PR are the tangents to the circle.

Join OQ and OR.

∠PQO is an angle in the semi-circle
of the blue circle
And we know that angle in a
semi-circle is a right angle.

∴ ∠PQO = 90°

⇒ OQ ⊥ PQ

Since OQ is the radius of the circle,

PQ has to be a tangent of the circle.

Similarly, PR is a tangent of the circle.


  1. Chapter 11 Class 10 Constructions
  2. Serial order wise
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Transcript

To construct the tangents to a circle from a point outside it. We are given a circle with centre O and a point P outside it. We have to construct the two tangents from P to the circle. Steps of Construction: Join PO and bisect it. Let M be the midpoint of PO. Taking M as centre and MO as radius, draw a circle. Let it  intersect the given circle at the points Q and R. Join PQ and PR. Then, PQ and PR are the required two tangents. Justification: We need to prove that PQ and PR are the tangents to the circle. Join OQ and OR. ∠PQO is an angle in the semi-circle of the blue circle And we know that angle in a semi-circle is a right angle. ∴ ∠PQO = 90° ⇒ OQ ⊥ PQ Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle.

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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