Perpendicular at contact point to tangent passes through circle center

Perpendicular lines are the two distinct lines that intersect at each other at $90^{\circ}$. Have you noticed anything common between the joining corners of your walls or the letter "L"? They are the straight lines known as perpendicular lines that meet each other at a specific angle - the right angle. We say that a line is perpendicular to another line if the two lines meet at an angle of $90^{\circ}$. The word "tangent" means "to touch". The Latin word for the same is "tangere". In general, we can say that the line that intersects the circle exactly at one point on its circumference and never enters the circle's interior is a tangent. A circle can have many tangents. They are perpendicular to the radius. In this article, we will see the proof of the perpendicular at the point of contact to the tangent.

What is Perpendicular?

A perpendicular is a straight line that makes an angle of $90^{\circ}$ with another line. $90^{\circ}$ is also called a right angle and is marked by a little square between two perpendicular lines as shown in the figure. Here, the two lines intersect at a right angle, and hence, are said to be perpendicular to each other.

Perpendicular line

What is a Tangent?

A tangent is a line that just barely touches a curve at a single point, without actually intersecting the curve. In other words, it's a line that's coincident to the curve at that particular point. The point where it touches the edge of the circle is called the point of tangency.

Line t is tangent to the circle in the figure.

Prove the Perpendicular at the Point of Contact to the Tangent that Passes Through the Centre of the Circle

The perpendicular at the point of contact to the tangent to a circle passes via the centre of the circle. This can be proven by considering the tangent to the circle as a line that is perpendicular to the radius at the point of contact. Therefore, the perpendicular bisector of the radius at the point of contact passes through the centre of the circle.

Construction

Let's draw a tangent PQ to a circle as shown below.

Example

As we know, a tangent at any point of a circle is perpendicular to the radius through the point of contact.

At the point of contact P, RP is perpendicular to the tangent PQ.

We also know that the radius or diameter will always pass through the centre of the circle.

Therefore, PR passes through the centre O.

Hence it is proved that perpendicular PR of tangent PQ passes through centre O.

Solved Examples

Q 1. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. What will be the radius of the circle?

Ans: So first of all we are going to draw the figure based on the information we have.

Tangent PQ

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

Therefore, OPQ is a right-angled triangle.

By Pythagoras theorem,

$O Q^2=O P^2+P Q^2$

$25^2=r^2+24^2$

$r^2=25^2-24^2$

$r^2=625-576$

$r^2=49$

$r=\pm 7$

Radius cannot be a negative value, hence, $r=7 \mathrm{~cm}$.

Q 2. What will be the length of the tangent drawn from a point 8 cm away from the centre of a circle of radius 6 cm?

Ans: Let $\mathrm{O}$ be the circle's centre. $\mathrm{OA}$ is the radius of the circle and $\mathrm{OP}$ is $8 \mathrm{~cm}$ According to question, we have

$\angle \mathrm{OAP}=90^{\circ}$

So, by Pythagoras theorem, we get

$\mathrm{OP}^2=\mathrm{OA}^2+\mathrm{AP}^2$

$\Rightarrow \mathrm{AP}^2=\mathrm{OP}^2-\mathrm{OA}^2$

$\Rightarrow \mathrm{AP}^2=8^2-6^2$

$\Rightarrow \mathrm{AP}^2=64-36$

$\Rightarrow \mathrm{AP}^2=28$

$\Rightarrow \mathrm{AP}=2 \sqrt{7} \mathrm{~cm}$

Therefore, the length of the tangent will become $2 \sqrt{7} \mathrm{~cm}$.

Practice Questions

Q 1. Perpendicular lines ___________

1. will never intersect

2. intersect to form right angles

3. are curved lines

4. are not coplanar

Ans: Intersect to form right angles

Q 2. If the length of the tangent line is 4 cm drawn from any point, 10 cm away from the centre of the circle, then what will be the radius of that circle?

Ans: $\sqrt{84} \mathrm{~cm}$

Summary

A perpendicular makes a 90 degree angle. A tangent touches the circle at one point only which is called the tangency point.The perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle. This can be proven by considering the tangent to the circle as a line that is perpendicular to the radius at the point of contact. Therefore, the perpendicular bisector of the radius at the point of contact passes through the centre of the circle.