Learning Points
1.Tangent
2. Point of contact
3. Theorem 10.1
4. Theorem 10.2
Tangent
A line which passes through a certain point of a circle is said to be tangent whereas the point is said to be point of contact.
Theorem 10.1
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Given
There is a circle of centre O. A tangent XY passes through the point P. OP is the radius through the point of contact P.
Prove that
XY perpendicular on OP.
Construction
Take a point Q on XY.
Proof
We know that the shortest distance between a point and line is perpendicular to the line.
For all points on XY, OP is the shortest distance between O and XY.
So, XY is perpendicular on the radius OP.
Proved.
Theorem 10.2
The lengths of tangents drawn from an external point to a circle are equal.
Solution:
Given:
PQ and PR are the tangents from the external point P.
To Prove that:
PQ = PR
Construction:
Join O to P, Q and R.
Proof:
Consider in ∆OPQ and ∆OPR.
OP = OP (Common)
OQ = OR (Radius of same circle)
<OQP = <ORP (Each right angle)
∆OPQ ≈ ∆OPR (RHS)
Therefore,
PQ = PR Proved.
