Here we will prove that the tangent at any point of a circle and the radius through the point are perpendicular to each other.
Given:
A circle with centre O in which OP is a radius. XPY is a tangent drawn to the circle at the point P.
To prove: OP ⊥ XY.
Construction: On XY take any point Q, other than P. Join O to Q.
Proof:
Statement |
Reason |
1. OQ > OP ⟹ OP is the shortest line segment that can be drawn to the tangent XY from O. |
1. Every point on the tangent, other than P, lies outside the circle. |
2. OP ⊥ XY. (Proved) |
2. The shortest line segment, drawn to a line from a point outside it, is perpendicular to the line. |
Note:
1. One and only one tangent can be drawn to a circle at a given point on the circumference because only one perpendicular can be drawn to Op through the point P.
2. The perpendicular to a tangent through its point of contact passes through centre of the circle because only one perpendicular, OP, can be drawn to the line XY through the point P.
3. The radius drawn perpendicular to the tangent passes through the point of contact because only one perpendicular, Op can be drawn to XY from the point O.
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