# Tangent - Perpendicular to Radius

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.

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Relation between Diameter Radius and Circumference |Problems |Examples

Apr 22, 24 05:19 PM

Relation between diameter radius and circumference are discussed here. Relation between Diameter and Radius: What is the relation between diameter and radius? Solution: Diameter of a circle is twice

2. ### Circle Math | Terms Related to the Circle | Symbol of Circle O | Math

Apr 22, 24 01:35 PM

In circle math the terms related to the circle are discussed here. A circle is such a closed curve whose every point is equidistant from a fixed point called its centre. The symbol of circle is O. We…

3. ### Preschool Math Activities | Colorful Preschool Worksheets | Lesson

Apr 21, 24 10:57 AM

Preschool math activities are designed to help the preschoolers to recognize the numbers and the beginning of counting. We believe that young children learn through play and from engaging

4. ### Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

Apr 20, 24 05:39 PM

There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…