Theorems on Circle
Tangent to a circle: A line which intersects a circle at any one point is called the tangent.
- There is only one tangent at a point of the circle.
- The tangent to a circle is perpendicular to the radius through the point of contact.
- The lengths of the two tangents from an external point to a circle are equal.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Construction: Draw a circle with centre O. Draw a tangent XY which touches point P at the circle.
To Prove: OP is perpendicular to XY.
Draw a point Q on XY; other than O and join OQ. Here OQ is longer than the radius OP.
OQ > OP
For every point on the line XY other than O, like Q1, Q2, Q3, ……….Qn;
`OQ_1 > OP`
`OQ_3 > OP`
`OQ_4 > OP`
Since OP is the shortest line
Hence, OP ⊥ XY proved
The lengths of tangents drawn from an external point to a circle are equal.
Construction: Draw a circle with centre O. From a point P outside the circle, draw two tangents P and R.
To Prove: PQ = PR
Proof: In Δ POQ and Δ POR
`OQ = OR` (radii)
`PO = PO` (common side)
`∠PQO = ∠PRO` (Right angle)
Hence; `Δ POQ ≅ Δ POR` proved