Introduction to Perpendicular and Tangent
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 ___________
will never intersect
intersect to form right angles
are curved lines
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.
FAQs on Perpendicular at contact point to tangent passes through circle center
1. What is the "point of contact" in relation to a circle's tangent?
The point of contact is the single, unique point where a tangent line touches a circle's circumference. At this specific point, the line does not cross into the circle's interior but simply grazes its edge. All geometric properties, such as the perpendicular relationship with the radius, are defined at this exact point.
2. What is the fundamental relationship between a circle's radius and its tangent?
The fundamental relationship, which is a key theorem in geometry, states that the radius of a circle is always perpendicular to the tangent at the point of contact. This means the angle formed between the radius (when drawn to the point of contact) and the tangent line is exactly 90 degrees.
3. How can we prove that the perpendicular to a tangent at the point of contact must pass through the circle's centre?
The proof for this theorem typically uses the method of contradiction, as per the NCERT syllabus for the 2025-26 session. The steps are:
Assume a perpendicular line is drawn from the point of contact 'P' to the tangent, and it passes through a point 'Q' which is *not* the centre 'O'.
By this assumption, the angle ∠QPT would be 90°.
However, we know from the base theorem that the radius is perpendicular to the tangent (OP ⊥ PT), which means ∠OPT must be 90°.
This leads to a contradiction: ∠QPT cannot equal ∠OPT if Q and O are different points. Therefore, our initial assumption was false, and the perpendicular line must pass through the centre 'O'.
4. Why is it impossible for the perpendicular from the point of contact to pass through any point other than the circle's centre?
It is impossible because the shortest distance from a point (the centre) to a line (the tangent) is always the perpendicular distance. The radius is, by definition, this shortest perpendicular line from the centre to the tangent at the point of contact. If another line from the point of contact were also perpendicular to the tangent but didn't go through the centre, it would imply that two different lines from the same point could be perpendicular to the same tangent line. This violates a fundamental principle of geometry which states that a line can only have one unique perpendicular at a given point.
5. What is the main difference between the theorem 'radius is perpendicular to the tangent' and 'the perpendicular at the contact point passes through the centre'?
These two theorems describe the same geometric fact but from different starting points; one is essentially the converse of the other. The key difference is the focus:
Radius ⊥ Tangent: This theorem starts by identifying the radius and concludes that it forms a 90° angle with the tangent. Its purpose is to establish the perpendicularity.
Perpendicular Passes Through Centre: This theorem starts with a line that is already known to be perpendicular to the tangent at the contact point and concludes that this line must pass through the centre. Its purpose is to locate the centre.
6. How is this theorem applied when solving geometry problems?
This theorem is crucial for solving problems where the centre of a circle is an unknown that needs to be found. If you can identify a tangent line and its point of contact, you can construct a perpendicular line from that point. The theorem guarantees that the centre of the circle lies somewhere on this constructed line, which significantly narrows down its possible location and helps in finding coordinates or proving other geometric properties.
7. What is a common mistake to avoid when using this theorem?
A very common mistake is assuming that any line drawn from the centre to the tangent is perpendicular. The 90° relationship holds true only at the exact point of contact. A line segment from the centre to any other point on the tangent will be the hypotenuse of a right-angled triangle and will always be longer than the radius. Always identify the precise point of contact before applying the perpendicularity rule.
8. How many tangents can be drawn to a circle at a single point on its circumference?
Only one and only one tangent can be drawn at any single point on a circle. If you were to draw another distinct line through that same point, it would inevitably cross into the interior of the circle, making it a secant, not a tangent.
