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.
From Tangent - Perpendicular to Radius to HOME PAGE
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
-
Patterns in Numbers | Patterns in Maths |Math Patterns|Series Patterns
Dec 13, 24 08:43 AM
We see so many patterns around us in our daily life. We know that a pattern is an arrangement of objects, colors, or numbers placed in a certain order. Some patterns neither grow nor reduce but only r… -
Patterns in Math | Missing Number | Counting Numbers | Worksheets
Dec 13, 24 12:31 AM
Finding patterns in math is very important to understand the sequence in the series. We need to find the exact missing number that from the group of numbers. The counting numbers may be counting -
Concept of Pattern | Similar Patterns in Mathematics | Similar Pattern
Dec 12, 24 11:22 PM
Concept of pattern will help us to learn the basic number patterns and table patterns. Animals such as all cows, all lions, all dogs and all other animals have dissimilar features. All mangoes have si… -
2nd Grade Geometry Worksheet | Plane and Solid Shapes | Point | Line
Dec 12, 24 10:31 PM
2nd grade geometry worksheet -
Types of Lines |Straight Lines|Curved Lines|Horizontal Lines| Vertical
Dec 09, 24 10:39 PM
What are the different types of lines? There are two different kinds of lines. (i) Straight line and (ii) Curved line. There are three different types of straight lines. (i) Horizontal lines, (ii) Ver…
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.