The tangent at any point of a circle is perpendicular to the radius through the point of contact

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Given: A circle with center O, and a tangent XY to the circle at a point P.

Prove: OP ⟂ XY

Proof : Let Q be any point on XY other than P and join OQ.

The point Q must lie outside the circle, because if Q lies inside the circle, XY will become a secant and not a tangent to the circle.

Therefore, OQ is longer than the radius OP of the circle.
that is OQ > OP as OP = OR

Since this happens for every point on the line XY except the point P, OP is the shortest of all the distances of the point O to the points of XY.

So, OP is perpendicular to XY.

OP ⟂ XY

Hence Proved

Related posts:

  1. Theorem – The perpendicular from the center of a circle to a chord bisects the chord
  2. Tangent of a Circle Definition & Example
  3. Theorem – The perpendicular from the center of a circle to a chord bisects the chord
  4. The angle subtended by an arc at the centre is double of the angle subtended by it at any point on the remaining part of the circle – Theorem – geometry
  5. Prove that the angle in a semicircle is a right angle – Theorem – Geometry
  6. Draw a Circle of a Given Radius – Using Compass
  7. Cyclic Quadrilaterals: Definition, Properties, Examples
  8. Theorem – If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal
  9. Theorem – Equal chords of a circle subtend equal angles at the center
  10. Radius of a Circle

Leave a Reply

Your email address will not be published. Required fields are marked *