# What are the 3 key theorems about a tangent to a circle?

### Today in this article I will explain a very important topics,which is circle and its various theorems.

## Contents:-

## *Prove that radius of a circle is perpendicular on the tangent at its point of contact.

### Answer:-

### To prove:- OR⊥PQ

### Construction:- 'M' is an another point taken on 'PQ' outside the circle

### Proof:-

If M would inside the circle then PQ would be a secant not a tangent, but in this theorem PQ must be a tangent, So clearly we can see that the distance of 'M' from the center of the circle 'O' is greater than the distance of point 'R' from the center 'O'

###

i.e; OM> OR, and this is universal true for any pont lying outside the circle and which is on PQ. So 'OR' is the least smaller distance from center 'O'

∵ We know that line drawn from a fixed point on another line the shortest vertical line is the the perpendicular.

So, OR⊥ PQ Proved.

## *Prove that the length of the tangents drawn from an external points on a circle are equal and they also produces equal angles at the center.

### Answer:-

### Let P be an external point, PA and PB are two drawn tangents on the circle 'O' at the points 'A' and 'B' respectively.

### To prove:-

(i) PA = PB

(ii) <POA = <POB

Construction:- 'OP' is joined.

Proof:- ∵ OA is the radius of the circle and PA is a tangent at point A of the circle,

∴ OA⊥ PA

∴ ∠OAP = 90°

Similarly,

<OBP = 90°

Now,

In between Δs AOP and BOP we get,

∵ ∠OAP=∠OBP [ ∵ Both are 90°]

∵ OA = OB [ ∵ Radii of the same circle]

and,

OP = OP[∵Common side of the Δs]

∴ Δ AOP ≌ Δ BOP

∴ AP = BP

## * What is a Cyclic quadrilateral?

### Answer:-

### A quadrilateral whose all four angular points touches the arc of a circle is called a Cyclic quadrilateral.

## * Prove that sum of opposite angles of a cyclic quadrilateral is 180°

### Answer:-

### Let 'ABCD' is a cyclic quadrilateral.

To prove:- <ABC + <ADC = 180°

and <BAD + < BCD = 180°

Construction:- OA and OC are joined.

Proof:- ∵∠ABC and obtuse ∠AOC are the angles at the circumference and at the center standing on the same arc ADC

∴∠ABC = ½ obtuse <AOC…..(i) [∵angle at the circumference is half the angle at the center standing on the same arc]

Again,

<ADC and reflex <AOC are the angles at the circumference and at the center standing on the same arc ABC

∴ ∠ADC =½ reflex <AOC…..(2) [angle at the circumference is half the angle at the center standing on the same arc]

(1)+(2)

<ABC+<ADC=½(obtuse<AOC+reflex<AOC)

=><ABC+<ADC=½ ×360°

=><ABC+<ADC =180° Proved.

## *Prove that semi circle angle is 90°

### Answer:-

### Let, AB is a diameter of a circle with center'O' ∴∠ACB is an angle of semi circle

To prove:- <ACB = 90°

Proof:-

∵∠ACB and ∠AOB are the angles at the circumference and at the center respectively standing on the same arc AB

###

∴ ∠ACB = ½ <AOB [∵ angle at the circumference is half the angle at the center standing on the same arc]

=> <ACB = ½ ×180° [ ∵∠AOB is a straight angle.∴∠AOB=180°]

=><ACB = 90° Proved.

## *Prove that angles at the circumference of a circle are equal.

### Answer:-

### Let,

<ACB and <ADB are the angles at the circumference and <AOB at the center standing on the same arc APB

To prove:- <ACB = <ADB

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## FREQUENTLY ASKED QUESTIONS:-
##
How many tangents can be drawn in a circle?

Answer:- Infinite number of tangents can be drawn in a circle.

## What is the relation between a tangent and a radius of a circle ?

Answer :- Relation is = r ⊥ t where, r = radius of the circle and t = tangent of the circle.

## What is tangent of a circle?

Answer :- Any line which touches the circle at a point, then that line is called the tangent of that circle.

## In how many points a tangent touches a circle?

Answer:- One.

##
What is secent of a circle?

Answer:- Any line which touches the circle
at two different points, then that line is called the secent of that circle.

##
In how many points a secent touches a circle?

.
Answer:- Two

## How many tangents can be drawn in a circle?

Answer:- Infinite number of tangents can be drawn in a circle.

## What is the relation between a tangent and a radius of a circle ?

Answer :- Relation is = r ⊥ t where, r = radius of the circle and t = tangent of the circle.

## What is tangent of a circle?

Answer :- Any line which touches the circle at a point, then that line is called the tangent of that circle.

## In how many points a tangent touches a circle?

Answer:- One.

## What is secent of a circle?

Answer:- Any line which touches the circle at two different points, then that line is called the secent of that circle.

## In how many points a secent touches a circle?

. Answer:- Two