Properties of Tangents - Summary

Class 10Circles

This is a complete summary of all tangent properties and theorems covered in the Class 10 NCERT chapter on Circles. Tangent questions are among the most frequently tested topics in CBSE board examinations.

A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of contact (or point of tangency).

Understanding these properties thoroughly enables students to solve proof-based, calculation-based, and construction-based problems on tangents.

What is Properties of Tangents - Summary?

Key Definitions:

Tangent: A line that intersects a circle at exactly one point.
Point of Contact: The single point where the tangent touches the circle.
Secant: A line that intersects a circle at two points.
Normal: The line perpendicular to the tangent at the point of contact (passes through the centre).

Properties of Tangents - Summary Formula

All Tangent Theorems at a Glance:

#	Theorem	Statement
1	Tangent ⊥ Radius	The tangent at any point of a circle is perpendicular to the radius through the point of contact.
2	Equal Tangents	The lengths of tangents drawn from an external point to a circle are equal.
3	Centre-External Point Line	The line joining the centre to an external point bisects the angle between the two tangents from that point.
4	Tangent-Chord Angle	The angle between a tangent and a chord drawn from the point of contact equals the inscribed angle subtending the same arc on the opposite side.
5	Number of Tangents	From a point inside: 0 tangents. On the circle: 1 tangent. Outside: 2 tangents.

Length of tangent from external point:

Length = √(d² − r²)

where d = distance from external point to centre, r = radius.

Derivation and Proof

Proof: Tangent is perpendicular to radius (Theorem 1)

Let O be the centre, P be the point of contact, and l be the tangent at P.
Assume OP is NOT perpendicular to l. Then there exists a point Q on l such that OQ ⊥ l.
Since OQ ⊥ l, OQ < OP (perpendicular is the shortest distance).
But Q lies on the tangent (outside the circle), so OQ > r = OP. Contradiction.
Therefore, OP ⊥ l. The tangent is perpendicular to the radius at the point of contact.

Proof: Equal Tangents from External Point (Theorem 2)

Let PA and PB be tangents from external point P to circle with centre O.
In △OAP and △OBP:
OA = OB (radii)
OP = OP (common)
∠OAP = ∠OBP = 90° (Theorem 1)
By RHS congruence: △OAP ≅ △OBP
Therefore PA = PB (CPCT).

Types and Properties

Types of tangent problems in board exams:

Prove that tangent ⊥ radius: Direct application of Theorem 1.
Find tangent length: Use √(d² − r²).
Prove equal tangents: Apply Theorem 2 with RHS congruence.
Angle between tangents: If ∠APB = θ and PA, PB are tangents, then ∠AOB = 180° − θ.
Tangent from external point constructions: Draw tangents using geometric construction.
Mixed problems: Tangent + chord + inscribed angle combinations.
Quadrilateral with tangents: If tangents from vertices of a triangle to the incircle, use equal tangent property.

Solved Examples

Example 1: Length of Tangent from External Point

Problem: Find the length of the tangent from a point 13 cm away from the centre of a circle of radius 5 cm.

Solution:

d = 13 cm, r = 5 cm
Length = √(d² − r²) = √(169 − 25) = √144 = 12 cm

Answer: The tangent length is 12 cm.

Example 2: Angle Between Two Tangents

Problem: Two tangents PA and PB are drawn to a circle of radius 4 cm from a point P that is 8 cm from the centre. Find ∠APB.

Solution:

In △OAP: OA = 4, OP = 8, ∠OAP = 90°
sin(∠OPA) = OA/OP = 4/8 = 1/2
∠OPA = 30°
∠APB = 2 × ∠OPA = 60°

Answer: ∠APB = 60°.

Example 3: Tangents from Vertices of a Triangle

Problem: A circle is inscribed in △ABC with AB = 10 cm, BC = 8 cm, AC = 12 cm. Find the lengths of the tangent segments from each vertex.

Solution:

Let tangent lengths from A, B, C be x, y, z respectively.

x + y = AB = 10 ... (i)
y + z = BC = 8 ... (ii)
z + x = AC = 12 ... (iii)

Adding all: 2(x + y + z) = 30 → x + y + z = 15.

From (i): z = 15 − 10 = 5
From (ii): x = 15 − 8 = 7
From (iii): y = 15 − 12 = 3

Answer: Tangent lengths: from A = 7 cm, from B = 3 cm, from C = 5 cm.

Example 4: Proving ∠AOB + ∠APB = 180°

Problem: PA and PB are tangents from P to a circle with centre O. Prove that ∠AOB + ∠APB = 180°.

Solution:

In quadrilateral OAPB:

∠OAP = 90° (tangent ⊥ radius)
∠OBP = 90° (tangent ⊥ radius)
Sum of angles = 360°
∠OAP + ∠OBP + ∠AOB + ∠APB = 360°
90° + 90° + ∠AOB + ∠APB = 360°
∠AOB + ∠APB = 180°

Answer: ∠AOB + ∠APB = 180°. Hence proved.

Example 5: Tangent and a Chord

Problem: A tangent PQ at point P of a circle of radius 5 cm meets a line through the centre O at point Q such that OQ = 12 cm. Find PQ and the length of chord AB where AB passes through Q and is perpendicular to OQ.

Solution:

PQ = √(OQ² − OP²) = √(144 − 25) = √119 ≈ 10.9 cm

For chord AB ⊥ OQ through Q: Since this chord is outside the standard geometry (Q is external), this requires the secant-tangent relationship.

Answer: PQ = √119 cm ≈ 10.9 cm.

Example 6: Parallel Tangents

Problem: Prove that the tangents at the two ends of a diameter of a circle are parallel.

Solution:

Let AB be a diameter. Tangent at A = l₁, Tangent at B = l₂.
OA ⊥ l₁ (tangent ⊥ radius at A)
OB ⊥ l₂ (tangent ⊥ radius at B)
OA and OB are parts of the same line (the diameter AB)
So l₁ ⊥ AB and l₂ ⊥ AB
Two lines perpendicular to the same line are parallel

Answer: Tangents at the endpoints of a diameter are parallel. Hence proved.

Example 7: Incircle Perimeter Property

Problem: Prove that the perimeter of a triangle circumscribing a circle equals 2 × (sum of tangent lengths from any vertex)... Actually, prove that if a quadrilateral ABCD circumscribes a circle, then AB + CD = BC + DA.

Solution:

Let the tangent lengths from each vertex be:

From A: a (to two tangent points)
From B: b
From C: c
From D: d

Then:

AB = a + b, BC = b + c, CD = c + d, DA = d + a
AB + CD = (a + b) + (c + d)
BC + DA = (b + c) + (d + a)
Both equal a + b + c + d

Answer: AB + CD = BC + DA. Hence proved.

Real-World Applications

Properties of tangents are applied in:

Engineering: Designing gear teeth that are tangent to pitch circles.
Optics: Light rays tangent to curved surfaces determine reflection and refraction.
Road Design: Curves in roads are tangent to straight sections for smooth transitions.
Astronomy: Tangent lines determine the apparent size of celestial objects.
Architecture: Arched structures use tangent principles for smooth joins.

Key Points to Remember

A tangent touches the circle at exactly one point.
The tangent is perpendicular to the radius at the point of contact.
Two tangents from an external point are equal in length.
The angle between two tangents from an external point P and the angle subtended at the centre sum to 180°.
Length of tangent from external point = √(d² − r²).
From a point inside a circle, no tangent can be drawn.
From a point on the circle, exactly one tangent can be drawn.
From an external point, exactly two tangents can be drawn.
In a circumscribing quadrilateral: AB + CD = BC + DA.
The line joining the centre to an external point bisects the angle between the tangents.

Practice Problems

Find the length of the tangent from a point 10 cm away from the centre of a circle of radius 6 cm.
Two tangents from point P make an angle of 60°. If the radius is 5 cm, find the distance OP.
A circle is inscribed in a triangle with sides 6, 8, and 10 cm. Find the radius of the inscribed circle.
Prove that tangents at the endpoints of any chord make equal angles with the chord.
If PA and PB are tangents to a circle with ∠APB = 80°, find ∠AOB.
A quadrilateral ABCD circumscribes a circle. If AB = 6 cm, BC = 7 cm, CD = 4 cm, find DA.

Frequently Asked Questions

Q1. Why is the tangent perpendicular to the radius?

The radius to the point of contact is the shortest distance from the centre to the tangent line. The shortest distance from a point to a line is the perpendicular distance. Therefore, the radius is perpendicular to the tangent.

Q2. Can a tangent pass through the centre of a circle?

No. A tangent touches the circle at one point and is perpendicular to the radius at that point. It lies entirely outside the circle (except at the point of contact) and cannot pass through the centre.

Q3. What is the relationship between ∠APB and ∠AOB?

∠APB + ∠AOB = 180°. If the angle between the two tangents is 60°, then the angle subtended at the centre is 120°.

Q4. How many tangents can be drawn from a point inside the circle?

Zero. Every line through an interior point intersects the circle at two points (it is a secant), so no tangent exists from inside.

Q5. What is the tangent-tangent property for circumscribed quadrilaterals?

If a quadrilateral circumscribes a circle, the sum of opposite sides are equal: AB + CD = BC + DA. This follows from the equal tangent property applied at each vertex.

Q6. How do you find the tangent length without knowing the distance to the centre?

You need either the distance to the centre (d) and the radius (r), or additional geometric information (angles, other tangent lengths) to compute the tangent length.

Properties of Tangents - Summary

What is Properties of Tangents - Summary?

Properties of Tangents - Summary Formula

Derivation and Proof

Types and Properties

Solved Examples

Example 1: Length of Tangent from External Point

Example 2: Angle Between Two Tangents

Example 3: Tangents from Vertices of a Triangle

Example 4: Proving ∠AOB + ∠APB = 180°

Example 5: Tangent and a Chord

Example 6: Parallel Tangents

Example 7: Incircle Perimeter Property

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. Why is the tangent perpendicular to the radius?

Q2. Can a tangent pass through the centre of a circle?

Q3. What is the relationship between ∠APB and ∠AOB?

Q4. How many tangents can be drawn from a point inside the circle?

Q5. What is the tangent-tangent property for circumscribed quadrilaterals?

Q6. How do you find the tangent length without knowing the distance to the centre?

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