Theorem - The tangent at any point of a circle is perpendicular to the radius through the point of contact.
In the previous section, you have seen that a tangent to a circle is a line that intersects the circle at only one point.
1) A circle can only have one tangent at a point of the cirlce.
But the circle is made of infinite numbers of points therefore a circle can have infinitely many tangents. But it can only have one tangent at one point.
Theorem1- The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Given: A circle with centre O, a tangent XY at the point of contanct P.
To prove: OP ⊥ XY.
Construction: Take a point Q, other than P or XY. Jion OQ.
Proof: Q lies on the tangent XY.
Q lies outside the circle.
Let OQ cuts the circle at R.
OR < OQ (a part is less than a whole)
But, OR=OP (radii of the same circle)
So, OP < OQ
Thus, OP is shorter than any line segment joining O to any point on XY other than point P.
Therefore, OP is the shortest distance between O and line segment XY.
But, the shortest distance between a point and a line segment is the perpendicular distance.
Therefore, OP is perpendicular to XY.
Shaalaa.com | Theorem - Tangent at any point to the circle is perpendicular to the radius
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