How to Find the Tangent to a Circle and Solve Related Problems
Definition of Tangent to Circle
A line that joins two close points from a point on the circle is known as a tangent. In simple words, we can say that the lines that intersect the circle exactly in one single point are tangents. Only one tangent can be at a point to circle. The point where a tangent touches the circle is known as the point of tangency. The point where the circle and the line intersect is perpendicular to the radius. As it plays a vital role in the geometrical construction there are many theorems related to it which we will discuss further in this chapter.
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Here, point O is the radius, point P is the point of tangency.
Various Conditions of Tangency
Only when a line touches the curve at a single point it is considered a tangent. Or else it is considered only to be a line. Hence, we can define tangent based on the point of tangency and its position with respect to the circle.
When point lies on the circle
When point lies inside the circle
When point lies outside the circle
When Point Lies on the Circle
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Here, from the figure, it is stated that there is only one tangent to a circle through a point that lies on the circle.
When Point Lies Inside the Circle
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In the figure above, the point P is inside the circle. Now, all the lines passing through point P are intersecting the circle at two points. therefore, no tangent can be drawn to the circle that passes through a point lying inside the circle.
When Point Lies Outside the Circle
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The above figure concludes that from a point P that lies outside the circle, there are two tangents to a circle.
Properties of Tangent
Always remember the below points about the properties of a tangent
A line of tangent never crosses the circle or enters it; it only touches the circle.
The point at which the lien and circle intersect is perpendicular to the radius
The tangent segment to a circle is equal from the same external point.
A tangent and a chord forms an angle, the angle is exactly similar to the tangent inscribed on the opposite side of the chord.
Equation of Tangent to a Circle
Below is the equation of tangent to a circle
Tangent to a circle equation x2+ y2=a2 at (a cos θ, a sin θ) is x cos θ+y sin θ= a
Tangent to a circle equation x2+ y2=a2 at (x1, y1) is xx1+yy1= a2
Tangent to a circle equation x2+ y2=a2 for a line y = mx +c is y = mx ± a √[1+ m2]
Tangent to a circle equation x2+ y2=a2 at (x1, y1) is xx1+yy1= a2
Tangent to a Circle Formula
To understand the formula of the tangent look at the diagram given below.
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Here, we have a circle with P as its exterior point. From the exterior point P the circle has a tangent at Point Q and S. A straight line that cuts the curve in two or more parts is known as a secant. So, here the secant is PR and at point Q, R intersects the circle as shown in the diagram above. So, now we get the formula for tangent-secant
PR/PS = PS/ PQ
PS² = PQ.PR
Theorems of Tangents to Circle
Theorem 1
A radius is gained by joining the centre and the point of tangency. A tangent at the common point on the circle is at a right angle to the radius. The below diagram will explain the same where AB \[\perp\] OP
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Theorem 2
From one external point only two tangents are drawn to a circle that have equal tangent segments. A tangent segment is the line joining to the external point and the point of tangency. According to the below diagram AC = BC
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Examples of a Tangent to a Circle Formula
Example 1
In the below circle point O is the radius, PT is a tangent and OP is the radius, If PT is a tangent, then OP is perpendicular to PT.
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If OP = 3 Units and PT = 4 Units. Find the length of OT
Solution: as the radius is perpendicular to the tangent at the point of tangency, OP \[\perp\] PT
Therefore, ∠P is the right angle in the triangle OPT and triangle OPT is a right angle triangle.
Now, according to the Pythagoras theorem, we find OT.
(OP)² + (PT)² = (OT)²
3² + 4² = (OT)²
9 + 16 = (OT)²
25 = (OT)²
5 = OT
As the length cannot be negative, the length of OT is 5 units.
Example 2
In the below diagram PA and PB are tangents to the circle. Find the value of
∠OAP
∠AOB
∠OBA
∠ASB
The length of OP, PB = 7 cm (given)
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Solution:
∠OAP = 90° (Tangent is perpendicular to the radius)
∠AOB + ∠APB = 180°
∠AOB + 48° = 180°
∠AOB = 180° - 48° = 132°
∠OBA + ∠OAB + ∠AOB = 180° (angle sum of triangle)
2 x ∠OBA + ∠AOB = 180° (∠OBA = ∠OAB)
2 x ∠OBA + 132° = 180° (∠AOB = 132°)
∠OBA = 24°
∠AOB = 2 x ∠ASB (angle at centre = 2 angle at circle)
∠ASB = ∠AOB / 2
∠ASB = 132° / 2 = 66°
Cos 24° = \[\frac{7}{OP}\] ⇒ OP = \[\frac{7}{cos24^{0}}\]
FAQs on Tangent to a Circle: Meaning, Properties & Methods
1. What is a tangent to a circle?
A tangent is a straight line that touches a circle at exactly one point. This specific point is called the point of contact or point of tangency. The tangent line lies in the same plane as the circle and remains on the exterior, only grazing the circumference at that single point. It does not cross into the circle's interior.
2. What is the difference between a tangent and a secant of a circle?
The key difference lies in the number of points they intersect the circle:
- A tangent touches the circle at exactly one point (the point of contact).
- A secant is a line that intersects the circle at two distinct points, passing through the interior of the circle.
3. What are the two main theorems related to tangents in the CBSE Class 10 syllabus for 2025-26?
The two fundamental theorems for the chapter 'Tangent to a Circle' are:
- Theorem 1: The tangent at any point of a circle is perpendicular (at a 90° angle) to the radius through the point of contact.
- Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.
4. How is the length of a tangent from an external point to a circle calculated?
To calculate the length of a tangent, you can use the Pythagorean theorem. Imagine a right-angled triangle formed by:
- The radius of the circle to the point of contact.
- The tangent line from the external point to the point of contact.
- The line segment connecting the centre of the circle to the external point (this is the hypotenuse).
5. How many tangents can be drawn to a circle from a single point?
The number of tangents you can draw depends on the location of the point relative to the circle:
- If the point is inside the circle, you can draw zero tangents.
- If the point is on the circle, you can draw exactly one tangent.
- If the point is outside the circle, you can draw exactly two tangents.
6. Why must a tangent be perpendicular to the radius at the point of contact?
A tangent is perpendicular to the radius at the point of contact because the radius represents the shortest distance from the centre of the circle to any point on the tangent line. In geometry, the shortest distance from a point to a line is always the perpendicular distance. If the line connecting the centre to the point of contact were not perpendicular, another shorter path would exist, which would mean that point is not the true point of tangency, contradicting the definition.
7. What is the importance of the line that connects an external point to the centre of the circle?
This line is very important because it acts as an angle bisector. When two tangents are drawn from an external point to a circle, the line joining this point to the centre bisects two angles:
- The angle formed between the two tangents.
- The angle formed at the centre by the two radii to the points of contact.
8. Can you provide a real-world example of a tangent to a circle?
A great real-world example is a bicycle chain resting on its gear (sprocket). The straight sections of the chain are tangent to the circular gears. Where the chain first touches the gear and where it last touches it are the points of tangency. Another example is a straight road that just touches a circular roundabout at one point.
