Tangent to a Circle

Q: What is a tangent to a circle?

A tangent is a straight line that touches the circle at exactly one point. This point is called the point of tangency or point of contact.

Q: How many tangents can be drawn from a point outside a circle?

Exactly two tangents can be drawn from any point outside a circle. Both tangents have equal length.

Q: Why is the tangent perpendicular to the radius?

This is proved by contradiction. If the tangent were not perpendicular to the radius, there would be a shorter distance from the centre to the tangent line, meaning the line would enter the circle and become a secant (intersecting at two points), contradicting the definition of a tangent.

Q: What is the tangent-radius perpendicularity property?

At the point where a tangent touches a circle, the tangent is always at 90° to the radius drawn to that point. Mathematically, if OT is the radius and PT the tangent at T, then ∠OTP = 90°.

Q: Why are tangent lengths from an external point equal?

The two triangles formed (△OAP and △OBP) are congruent by RHS criterion: OA = OB (radii), OP = OP (common), and both have a 90° angle. By CPCT, PA = PB.

Q: What is a tangential quadrilateral?

A tangential quadrilateral is a quadrilateral that has an inscribed circle (a circle touching all four sides). For such a quadrilateral, the sum of opposite sides are equal: AB + CD = BC + DA.

Q: How do you find the tangent length from an external point?

Use the Pythagoras Theorem. If OP is the distance from the centre to the point, and r is the radius, then tangent length = √(OP² − r²). This works because the tangent, radius, and line from centre to the point form a right triangle.

Q: What is the difference between a tangent and a secant?

A tangent touches the circle at exactly one point. A secant intersects the circle at two points. As a secant moves outward and its two intersection points approach each other, it becomes a tangent in the limiting position.

Class 10Circles

A tangent to a circle is a straight line that touches the circle at exactly one point. This concept is introduced in CBSE Class 10 Mathematics, Chapter 10 (Circles).

Unlike a secant (which intersects a circle at two points), a tangent meets the circle at precisely one point called the point of tangency or point of contact. The properties of tangents form the basis for several important theorems and constructions in geometry.

Two fundamental results about tangents are proved in this chapter:

A tangent at any point of a circle is perpendicular to the radius through the point of contact.
The lengths of tangents drawn from an external point to a circle are equal.

What is Tangent to a Circle - Properties, Theorems & Solved Examples?

Definition: A tangent to a circle is a line that intersects the circle at exactly one point. This point is called the point of tangency (or point of contact).

Related terms:

Secant — a line that intersects the circle at two points.
Point of tangency (T) — the single point where the tangent touches the circle.
External point — a point outside the circle from which tangents can be drawn.
Internal point — a point inside the circle; no tangent can pass through it.

Number of tangents based on position of a point:

Position of Point	Number of Tangents
Inside the circle	0 (no tangent possible)
On the circle	1 (exactly one tangent)
Outside the circle	2 (two tangents)

Tangent to a Circle Formula

Theorem 1: Tangent-Radius Perpendicularity

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

If OT is the radius and PT is the tangent at T, then:

OT ⊥ PT (∠OTP = 90°)

Theorem 2: Equal Tangent Lengths

The lengths of tangents drawn from an external point to a circle are equal.

If PA and PB are tangents from external point P to a circle with centre O, touching at A and B, then:

PA = PB

Additional properties:

∠OAP = ∠OBP = 90° (tangent ⊥ radius)
OP bisects ∠APB (the angle between the two tangents)
OP bisects AB perpendicularly
∠APO = ∠BPO

Derivation and Proof

Proof of Theorem 1: Tangent is Perpendicular to Radius

To prove: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Let O be the centre, r the radius, and PT the tangent at point T.
Assume OT is NOT perpendicular to PT. Then there exists some other point Q on PT such that OQ ⊥ PT.
In right △OQT: OQ < OT (hypotenuse is the longest side).
So OQ < r. This means Q lies inside the circle.
But Q is on line PT. If Q is inside the circle, then PT enters the circle and must intersect it at two points — contradicting the fact that PT is a tangent (touches at only one point).
Therefore, our assumption is wrong. OT ⊥ PT. Hence proved.

Proof of Theorem 2: Equal Tangent Lengths from an External Point

To prove: PA = PB, where PA and PB are tangents from external point P to a circle with centre O.

Join OA, OB, and OP.
OA = OB = r (radii of the same circle).
∠OAP = ∠OBP = 90° (tangent ⊥ radius, by Theorem 1).
In △OAP and △OBP:
- OA = OB (radii)
- OP = OP (common side)
- ∠OAP = ∠OBP = 90°
By RHS congruence: △OAP ≅ △OBP.
Therefore PA = PB (CPCT). Hence proved.
Also by CPCT: ∠OPA = ∠OPB, so OP bisects ∠APB.

Types and Properties

Problems on tangents to circles in Class 10 include:

Type 1: Finding Tangent Length

Given radius, centre, and external point, find the tangent length using Pythagoras Theorem.
In right △OTP: PT² = OP² − OT² (since ∠OTP = 90°).

Type 2: Proving Properties Using Tangent Theorems

Prove that tangent segments from an external point are equal.
Prove angle bisector properties or perpendicularity results.

Type 3: Tangent and Secant Combinations

Problems involving both a tangent and a secant from the same point.

Type 4: Tangent to Inscribed/Circumscribed Figures

A circle inscribed in a triangle or quadrilateral. Use equal tangent lengths to find unknown sides.

Type 5: Finding Angles

Given tangent lines and radii, find angles using the perpendicularity property and angle sum properties.

Type 6: Two Tangents — Finding Angle Between Them

Find the angle between two tangents from an external point, given the angle subtended at the centre.

Methods

Method 1: Finding Tangent Length from an External Point

Join the centre O to the external point P (length = OP).
Join the centre O to the point of tangency T (length = OT = radius = r).
∠OTP = 90° (tangent ⊥ radius).
In right △OTP: PT = √(OP² − r²).

Method 2: Using Equal Tangent Lengths in Polygons

If a circle is inscribed in a quadrilateral ABCD, tangent points divide each side.
Mark tangent segments from each vertex: AP = AS, BP = BQ, CQ = CR, DR = DS (equal tangent lengths).
Add them: AB + CD = BC + DA. This is a standard result for tangential quadrilaterals.

Method 3: Finding Angle Between Two Tangents

Let PA and PB be tangents from P to a circle with centre O.
∠AOB + ∠APB = 180° (since OAPB is a cyclic quadrilateral with ∠A = ∠B = 90°).
So ∠APB = 180° − ∠AOB.

Key Facts for Problem Solving:

Tangent ⊥ radius at the point of contact — always gives a right angle.
Equal tangent lengths — PA = PB from any external point P.
OP bisects both ∠APB and chord AB.
In quadrilateral OAPB: ∠A + ∠B = 180°, so ∠O + ∠P = 180°.

Solved Examples

Example 1: Finding Tangent Length

Problem: A point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from P to the circle.

Solution:

Given:

OP = 13 cm (distance from centre to external point)
OT = 5 cm (radius)

Using Pythagoras Theorem in right △OTP:

PT² = OP² − OT²
PT² = 169 − 25 = 144
PT = 12 cm

Answer: The tangent length is 12 cm.

Example 2: Equal Tangent Lengths

Problem: Two tangents PA and PB are drawn from an external point P to a circle with centre O of radius 6 cm. If OP = 10 cm, find PA and PB.

Solution:

Given:

OA = OB = 6 cm (radii)
OP = 10 cm

In right △OAP:

PA² = OP² − OA² = 100 − 36 = 64
PA = 8 cm

Since tangent lengths from an external point are equal:

PB = PA = 8 cm

Answer: PA = PB = 8 cm.

Example 3: Finding the Angle Between Two Tangents

Problem: Two tangents are drawn from an external point P to a circle with centre O. If ∠AOB = 120°, find ∠APB.

Solution:

Given:

∠AOB = 120°
∠OAP = ∠OBP = 90° (tangent ⊥ radius)

In quadrilateral OAPB:

∠OAP + ∠APB + ∠PBO + ∠AOB = 360°
90° + ∠APB + 90° + 120° = 360°
∠APB = 360° − 300° = 60°

Answer: ∠APB = 60°.

Example 4: Circle Inscribed in a Triangle

Problem: A circle is inscribed in △ABC with AB = 10 cm, BC = 8 cm, CA = 12 cm. The circle touches AB at P, BC at Q, and CA at R. Find AP, BQ, and CR.

Solution:

Using equal tangent lengths from each vertex:

Let AP = AR = x (tangents from A)
Let BP = BQ = y (tangents from B)
Let CQ = CR = z (tangents from C)

Setting up equations from side lengths:

AP + PB = AB → x + y = 10 … (i)
BQ + QC = BC → y + z = 8 … (ii)
CR + RA = CA → z + x = 12 … (iii)

Adding all three:

2(x + y + z) = 30 → x + y + z = 15

Solving:

From x + y + z = 15 and y + z = 8: x = 7 → AP = 7 cm
From x + y = 10 and x = 7: y = 3 → BQ = 3 cm
From y + z = 8 and y = 3: z = 5 → CR = 5 cm

Answer: AP = 7 cm, BQ = 3 cm, CR = 5 cm.

Example 5: Tangential Quadrilateral Property

Problem: Prove that the sum of opposite sides of a quadrilateral circumscribing a circle are equal, i.e., AB + CD = BC + DA.

Solution:

Given: ABCD is a quadrilateral circumscribing a circle touching AB at P, BC at Q, CD at R, DA at S.

Using equal tangent lengths:

AP = AS (tangents from A)
BP = BQ (tangents from B)
CQ = CR (tangents from C)
DR = DS (tangents from D)

Adding:

AB + CD = (AP + BP) + (CR + DR)
= (AS + BQ) + (CQ + DS)
= (AS + DS) + (BQ + CQ)
= DA + BC

Therefore AB + CD = BC + DA. Hence proved.

Example 6: Finding Radius from Tangent and External Distance

Problem: The tangent from a point P to a circle is 24 cm and the distance of P from the centre is 25 cm. Find the radius of the circle.

Solution:

Given:

PT = 24 cm (tangent length)
OP = 25 cm (distance to centre)

Using Pythagoras Theorem (∠OTP = 90°):

OP² = OT² + PT²
625 = OT² + 576
OT² = 49
OT = 7 cm

Answer: The radius is 7 cm.

Example 7: Tangent and Secant from Same Point

Problem: From an external point P, a tangent PT and a secant PAB are drawn to a circle. If PT = 8 cm and PA = 4 cm, find PB.

Solution:

Given:

PT = 8 cm (tangent)
PA = 4 cm (nearer intersection)

Using the Tangent-Secant relation: PT² = PA × PB

64 = 4 × PB
PB = 16 cm

Answer: PB = 16 cm.

Example 8: Proving OP Bisects the Angle Between Tangents

Problem: Prove that the line joining the centre of a circle to an external point bisects the angle between the two tangents drawn from that point.

Solution:

Given: PA and PB are tangents from P. O is the centre. A and B are points of tangency.

To prove: ∠OPA = ∠OPB.

In △OAP and △OBP:
OA = OB (radii)
PA = PB (equal tangent lengths)
OP = OP (common)
By SSS congruence: △OAP ≅ △OBP
∠OPA = ∠OPB (CPCT)

Hence OP bisects ∠APB. Proved.

Real-World Applications

Engineering — Gear Design:

The teeth of interlocking gears are designed so that the contact between teeth occurs along tangent lines to the pitch circles. This ensures smooth power transmission.

Road Design:

When a straight road meets a curved section (roundabout or bend), the straight portion is tangent to the curve, ensuring a smooth transition for vehicles.

Optics:

Light rays tangent to a curved mirror or lens surface determine the angle of reflection or refraction at that point.

Architecture:

Tangent lines are used in designing arches, domes, and curved walls where straight structural elements meet curved surfaces.

Astronomy:

The concept of a tangent line to a planet's orbit is used to calculate escape velocity and trajectory angles.

Computer Graphics:

Bezier curves and splines use tangent vectors to control the smoothness and direction of curves.

Key Points to Remember

A tangent touches the circle at exactly one point (the point of tangency).
A tangent at any point is perpendicular to the radius at that point (∠OTP = 90°).
From an external point, exactly two tangents can be drawn. Their lengths are equal.
From a point on the circle, exactly one tangent can be drawn.
From a point inside the circle, no tangent can be drawn.
If PA and PB are tangents from P, then OP bisects ∠APB and AB perpendicularly.
In quadrilateral OAPB: ∠AOB + ∠APB = 180°.
For a tangential quadrilateral: AB + CD = BC + DA (sum of opposite sides are equal).
Tangent length from external point P: PT = √(OP² − r²).
The tangent-radius perpendicularity is proved by contradiction. The equal tangent lengths theorem uses RHS congruence.

Practice Problems

A point is 17 cm from the centre of a circle of radius 8 cm. Find the tangent length.
Two tangents from an external point P make an angle of 60° with each other. If the radius is 5 cm, find OP.
Prove that the tangent drawn at the midpoint of an arc is parallel to the chord joining the endpoints of the arc.
A circle inscribed in a right triangle with legs 6 cm and 8 cm. Find the radius of the inscribed circle.
AB and CD are two parallel tangents to a circle of radius 5 cm. Find the distance between AB and CD.
From an external point, two tangents are drawn to a circle of radius 7 cm. If the tangent length is 24 cm, find the distance of the external point from the centre.
If the angle between two tangents drawn from an external point is 80°, find the angle subtended by the line segment joining the points of tangency at the centre.
A circle is inscribed in a quadrilateral with sides 5 cm, 7 cm, 8 cm, and x cm. Find x.

Frequently Asked Questions

Q1. What is a tangent to a circle?

A tangent is a straight line that touches the circle at exactly one point. This point is called the point of tangency or point of contact.

Q2. How many tangents can be drawn from a point outside a circle?

Exactly two tangents can be drawn from any point outside a circle. Both tangents have equal length.

Q3. Why is the tangent perpendicular to the radius?

This is proved by contradiction. If the tangent were not perpendicular to the radius, there would be a shorter distance from the centre to the tangent line, meaning the line would enter the circle and become a secant (intersecting at two points), contradicting the definition of a tangent.

Q4. What is the tangent-radius perpendicularity property?

At the point where a tangent touches a circle, the tangent is always at 90° to the radius drawn to that point. Mathematically, if OT is the radius and PT the tangent at T, then ∠OTP = 90°.

Q5. Why are tangent lengths from an external point equal?

The two triangles formed (△OAP and △OBP) are congruent by RHS criterion: OA = OB (radii), OP = OP (common), and both have a 90° angle. By CPCT, PA = PB.

Q6. What is a tangential quadrilateral?

A tangential quadrilateral is a quadrilateral that has an inscribed circle (a circle touching all four sides). For such a quadrilateral, the sum of opposite sides are equal: AB + CD = BC + DA.

Q7. How do you find the tangent length from an external point?

Use the Pythagoras Theorem. If OP is the distance from the centre to the point, and r is the radius, then tangent length = √(OP² − r²). This works because the tangent, radius, and line from centre to the point form a right triangle.

Q8. What is the difference between a tangent and a secant?

A tangent touches the circle at exactly one point. A secant intersects the circle at two points. As a secant moves outward and its two intersection points approach each other, it becomes a tangent in the limiting position.

Tangent to a Circle

What is Tangent to a Circle - Properties, Theorems & Solved Examples?

Tangent to a Circle Formula

Derivation and Proof

Types and Properties

Methods

Solved Examples

Example 1: Finding Tangent Length

Example 2: Equal Tangent Lengths

Example 3: Finding the Angle Between Two Tangents

Example 4: Circle Inscribed in a Triangle

Example 5: Tangential Quadrilateral Property

Example 6: Finding Radius from Tangent and External Distance

Example 7: Tangent and Secant from Same Point

Example 8: Proving OP Bisects the Angle Between Tangents

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. What is a tangent to a circle?

Q2. How many tangents can be drawn from a point outside a circle?

Q3. Why is the tangent perpendicular to the radius?

Q4. What is the tangent-radius perpendicularity property?

Q5. Why are tangent lengths from an external point equal?

Q6. What is a tangential quadrilateral?

Q7. How do you find the tangent length from an external point?

Q8. What is the difference between a tangent and a secant?

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