To construct the tangents to a circle from a point outside it.
We are given a circle with center O
and point P outside the circle
We need to draw tangents from point P to the circle
Let’s follow these steps
Steps of construction
Make perpendicular bisector of PO
Let M be the midpoint of PO.
Note: To learn how to draw perpendicular bisector, check Construction 11.2 of Class 9
2.Taking M as centre and MO as radius, draw a circle.
3.Let it intersect the given circle at points Q and R.
4.Join PQ and PR.
Then, PQ and PR are the required two tangents.
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. (Since Tangent is perpendicular to radius)
Similarly, PR is a tangent of the circle.
If center 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.
3.Point of intersection of perpendicular bisectors is the center of circle
Then you could proceed as Construction 11.3.