Circles - Tangents to a circle, properties of tangents drawn from an external point, and geometric proofs

The study of circles forms a cornerstone of Euclidean plane geometry. While introductory geometry focuses on basic definitions such as radii, diameters, chords, and arcs, Class 10 Mathematics transitions into the formal analysis of lines interacting with circles. Among these interactions, the concept of a tangent is paramount.

Understanding tangents is not merely an academic exercise; it bridges pure geometry with trigonometry, coordinate geometry, and real-world calculus applications. Tangents govern how physical objects roll, how forces act along curved trajectories, and how visual boundaries are formed. This study guide provides a complete, rigorous, and student-friendly breakdown of Chapter 10 (Circles) as per the NCERT/CBSE syllabus, focusing on theoretical foundations, formal geometric proofs, solved textbook problems, and critical exam strategies.

1. Core Conceptual Breakdown

1.1 Positional Relationships Between a Line and a Circle

Consider a circle with center $O$ and radius $r$, and a straight line $AB$ lying in the same plane. There are exactly three mutually exclusive possibilities for their relative positions:

1. Non-Intersecting Line: The line $AB$ lies completely outside the circle and has no common points with it. The perpendicular distance from $O$ to $AB$ is strictly greater than the radius ($d > r$).
2. Secant: The line $AB$ intersects the circle at two distinct points, say $P$ and $Q$. The line cuts through the interior of the circle. The perpendicular distance from $O$ to $AB$ is strictly less than the radius ($d < r$).
3. Tangent: The line $AB$ touches the circle at exactly one point, say $P$. The line remains entirely outside the circle except at this single point of contact. The perpendicular distance from $O$ to $AB$ is exactly equal to the radius ($d = r$).

  Case 1: Non-Intersecting          Case 2: Secant             Case 3: Tangent
        (d > r)                        (d < r)                     (d = r)

          O                              O                            O
         /                              /|                           |
        /                              / | r                         | r
       /  r                           /  |                           |
      /                              P---|---Q                       P
  ----------- AB                     ----|---- AB              -------------- AB
                                         d                            (Point of Contact)

Definition Summary Table

1.2 Fundamental Property of Tangents (Theorem 10.1)

Theorem Statement

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

                  O
                 /|
                / |
               /  | r
              /   |
             Q----P------------ XY
                Point of
                Contact

Formal Geometric Proof

• Given: A circle $C(O, r)$ with center $O$ and radius $r$, and a line $XY$ which is tangent to the circle at point $P$.

• To Prove: $OP \perp XY$.

• Construction: Take a point $Q$ on $XY$ distinct from $P$. Join $OQ$. Let $OQ$ intersect the circle at point $R$.

• Proof:

  1. Since $XY$ is a tangent to the circle at $P$, $P$ is the only point on $XY$ that lies on the circle.
  2. Therefore, the point $Q$ must lie outside the circle.
  3. Since $Q$ lies outside the circle and $R$ lies on the circle, the distance $OQ$ must be greater than the radius $OR$. $OQ > OR$
  4. But $OR = OP$ (since both are radii of the same circle $C(O, r)$).
  5. Substituting $OP$ for $OR$, we get: $OQ > OP$
  6. Since $Q$ was chosen as an arbitrary point on $XY$ (other than $P$), this inequality holds true for every point on the line $XY$ except $P$.
  7. In geometry, the shortest line segment joining a given point ($O$) to a given line ($XY$) is the perpendicular segment.
  8. Since $OP$ is the shortest distance from $O$ to the line $XY$, it follows that: $OP \perp XY$

Hence Proved.

1.3 Tangents from an External Point (Theorem 10.2)

Number of Tangents From Various Points

• Point inside the circle: Zero tangents can be drawn (any line passing through an interior point will be a secant intersecting at two points).
• Point on the circle: Exactly one tangent can be drawn.
• Point outside the circle: Exactly two tangents can be drawn to the circle.

                  A
                 /|
                / |
               /  |
              /   |
             /    |
            P-----O
             \    |
              \   |
               \  |
                \ |
                  B

Definition: Length of a Tangent

The length of the line segment from the external point to the point of contact is called the length of the tangent. In the figure above, $PA$ and $PB$ are the lengths of the tangents from external point $P$.

Theorem Statement

Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

Formal Geometric Proof

• Given: A circle $C(O, r)$ with center $O$, an external point $P$, and two tangents $PA$ and $PB$ touching the circle at points $A$ and $B$, respectively.

• To Prove: $PA = PB$.

• Construction: Join $OA$, $OB$, and $OP$.

• Proof:

  1. Consider triangles $\Delta OAP$ and $\Delta OBP$.
  2. According to Theorem 10.1, the radius drawn to the point of contact is perpendicular to the tangent. $\angle OAP = 90^\circ \quad \text{and} \quad \angle OBP = 90^\circ$ Thus, $\Delta OAP$ and $\Delta OBP$ are right-angled triangles.
  3. In right triangles $\Delta OAP$ and $\Delta OBP$:
     • $\angle OAP = \angle OBP = 90^\circ$ (Right angle)
     • $OP = OP$ (Common hypotenuse)
     • $OA = OB$ (Radii of the same circle)
  4. By the RHS (Right Angle-Hypotenuse-Side) Congruence Criterion: $\Delta OAP \cong \Delta OBP$
  5. Corresponding Parts of Congruent Triangles (CPCT) are equal. Therefore: $PA = PB$

Hence Proved.

1.4 Critical Corollaries Derived from Theorem 10.2

From the congruence $\Delta OAP \cong \Delta OBP$, we immediately obtain three vital properties that form the basis of most Class 10 board exam problems:

1. Equal Angles at the Center: $\angle AOP = \angle BOP$ The line segment joining the external point to the center bisects the angle subtended by the radii at the center.

2. Equal Inclination of Tangents: $\angle APO = \angle BPO$ The line segment joining the external point to the center bisects the angle between the two tangents.

3. Supplementary Angle Relationship: In quadrilateral $OAPB$, the sum of interior angles is $360^\circ$: $\angle OAP + \angle APB + \angle OBP + \angle AOB = 360^\circ$ Substituting $\angle OAP = 90^\circ$ and $\angle OBP = 90^\circ$: $90^\circ + \angle APB + 90^\circ + \angle AOB = 360^\circ \implies \angle APB + \angle AOB = 180^\circ$ The angle between two tangents drawn from an external point is supplementary to the angle subtended by the line segments joining the points of contact at the center.

4. Circumscribing Quadrilateral Property: If a quadrilateral $ABCD$ circumscribes a circle (all four sides touch the circle), then the sum of opposite sides is equal: $AB + CD = AD + BC$

2. Real-World Applications

2.1 Mechanical Belt and Pulley Systems

In mechanical engineering, a continuous belt passing over two circular pulleys moves along tangent lines. The point where the belt leaves contact with the pulley wheel represents the exact point of contact of a tangent line. Designing optimal belt tension and calculating required belt lengths rely directly on tangent properties and right-triangle geometry.

       Pulley 1                             Pulley 2
        .---.                                .---.
      /       \   <--- Tangent Belt --->   /       \
     |    O1   |==========================|    O2   |
      \       /                            \       /
        '---'                                '---'

2.2 Vehicle Wheels and Mudguard Design

When a bicycle or car wheel rotates rapidly on a flat road (which acts as a ground-level tangent line to the circular wheel), mud particles sticking to the tire detach due to inertia. The path taken by these flying particles follows a line tangential to the circular tire at the point of detachment. Bicycle mudguards are geometrically extended along these tangent paths to capture debris effectively.

2.3 Satellite Line-of-Sight and Horizon Calculations

For telecommunication and satellite systems, the maximum distance a line-of-sight signal can travel to the surface of the Earth depends on tangents. A signal emitted from a satellite at an external point $P$ strikes the spherical surface of the Earth at tangent points $A$ and $B$. The geometric region enclosed between these tangent points defines the satellite's coverage zone.

3. Step-by-Step Solved Textbook Examples

Example 1: Calculating Tangent Length via Pythagoras Theorem

Problem: A point $P$ is at a distance of $25\text{ cm}$ from the center $O$ of a circle. The length of the tangent $PT$ drawn from $P$ to the circle is $24\text{ cm}$. Find the radius of the circle.

                  T
                 /|
              r / |
               /  | 24 cm
              /   |
             O----P
              25 cm

Solution:

• Step 1: Identify the geometric relationship. Let $T$ be the point of contact of the tangent line drawn from $P$. By Theorem 10.1, the radius $OT$ is perpendicular to the tangent $PT$ at the point of contact $T$. $\angle OTP = 90^\circ$

• Step 2: Apply the Pythagorean Theorem. In right-angled triangle $\Delta OTP$, $OP$ is the hypotenuse. $OP^2 = OT^2 + PT^2$

• Step 3: Substitute given values and solve. Given: $OP = 25\text{ cm}$, $PT = 24\text{ cm}$, $OT = r$. $25^2 = r^2 + 24^2$ $625 = r^2 + 576$ $r^2 = 625 - 576$ $r^2 = 49$ $r = \sqrt{49} = 7\text{ cm}$

Final Answer: The radius of the circle is $\mathbf{7\text{ cm}}$.

Example 2: Concentric Circles and Chord Bisector

Problem: Two concentric circles are of radii $5\text{ cm}$ and $3\text{ cm}$. Find the length of the chord of the larger circle which touches the smaller circle.

                     .---.
                   /   |   \
                  /    |    \
                 /     |5    \
                /      O      \
               /     / |       \
              /   3 /  |        \
             A-----P---|---------B
                   |---|
                     P

Solution:

• Step 1: Set up the geometric diagram and notation. Let $C_1$ be the larger circle with radius $R = 5\text{ cm}$ and $C_2$ be the smaller circle with radius $r = 3\text{ cm}$, both centered at $O$. Let $AB$ be a chord of $C_1$ that touches $C_2$ at point $P$.

• Step 2: Apply tangent properties to the inner circle $C_2$. Since $AB$ is tangent to $C_2$ at $P$, by Theorem 10.1: $OP \perp AB$

• Step 3: Apply chord properties to the outer circle $C_1$. In circle $C_1$, $AB$ is a chord and $OP \perp AB$. By the standard geometric theorem (a perpendicular drawn from the center of a circle to a chord bisects the chord): $AP = PB \implies AB = 2 \cdot AP$

• Step 4: Use right triangle $\Delta OPA$ to find $AP$. Join $OA$. In right-angled triangle $\Delta OPA$: $OA^2 = OP^2 + AP^2$ Substitute $OA = 5\text{ cm}$ and $OP = 3\text{ cm}$: $5^2 = 3^2 + AP^2$ $25 = 9 + AP^2$ $AP^2 = 16 \implies AP = 4\text{ cm}$

• Step 5: Calculate total chord length. $AB = 2 \cdot AP = 2 \times 4\text{ cm} = 8\text{ cm}$

Final Answer: The length of the chord of the larger circle is $\mathbf{8\text{ cm}}$.

Example 3: Circumscribing Quadrilateral Proof

Problem: Prove that a quadrilateral $ABCD$ circumscribing a circle satisfies $AB + CD = AD + BC$.

               D------R------C
              /   .-------.   \
             /  /           \  \
            S  |      O      |  Q
             \  \           /  /
              \   '-------'   /
               A------P------B

Solution:

• Step 1: State the Given and To Prove.

  • Given: Quadrilateral $ABCD$ circumscribes a circle centered at $O$, touching sides $AB, BC, CD,$ and $DA$ at points $P, Q, R,$ and $S$ respectively.
  • To Prove: $AB + CD = AD + BC$.

• Step 2: Apply Theorem 10.2 to all vertices. Lengths of tangents drawn from an external point to a circle are equal.

  • From vertex $A$: $AP = AS$ --- (Equation 1)
  • From vertex $B$: $BP = BQ$ --- (Equation 2)
  • From vertex $C$: $CR = CQ$ --- (Equation 3)
  • From vertex $D$: $DR = DS$ --- (Equation 4)

• Step 3: Add Equations (1), (2), (3), and (4). $(AP + BP) + (CR + DR) = (AS + BQ) + (CQ + DS)$

• Step 4: Regroup the segments based on quadrilateral sides. $(AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)$ Observing the diagram:

  • $AP + BP = AB$
  • $CR + DR = CD$
  • $AS + DS = AD$
  • $BQ + CQ = BC$

  Substituting these gives: $AB + CD = AD + BC$

Hence Proved.

Example 4: Advanced Angle Proof

Problem: Two tangents $TP$ and $TQ$ are drawn to a circle with center $O$ from an external point $T$. Prove that $\angle PTQ = 2 \angle OPQ$.

                  P
                 / \
                /   \
               /     \
              O-------T
               \     /
                \   /
                 \ /
                  Q

Solution:

• Step 1: Define variables for convenience. Let $\angle PTQ = \theta$.

• Step 2: Analyze triangle $\Delta TPQ$. By Theorem 10.2, $TP = TQ$. Therefore, $\Delta TPQ$ is an isosceles triangle with $TP = TQ$. $\angle TPQ = \angle TQP$

• Step 3: Use the angle sum property of $\Delta TPQ$. $\angle PTQ + \angle TPQ + \angle TQP = 180^\circ$ $\theta + 2\angle TPQ = 180^\circ$ $2\angle TPQ = 180^\circ - \theta$ $\angle TPQ = \frac{1}{2}(180^\circ - \theta) = 90^\circ - \frac{\theta}{2} \quad \text{--- (Equation 1)}$

• Step 4: Apply Theorem 10.1 to find $\angle OPT$. The radius $OP$ is perpendicular to tangent $TP$ at point $P$. $\angle OPT = 90^\circ$

• Step 5: Relate $\angle OPQ$ to $\angle OPT$ and $\angle TPQ$. From the figure: $\angle OPQ = \angle OPT - \angle TPQ$ $\angle OPQ = 90^\circ - \left(90^\circ - \frac{\theta}{2}\right)$ $\angle OPQ = 90^\circ - 90^\circ + \frac{\theta}{2}$ $\angle OPQ = \frac{\theta}{2}$ $2\angle OPQ = \theta$

• Step 6: Replace $\theta$ with $\angle PTQ$. $\angle PTQ = 2\angle OPQ$

Hence Proved.

4. Common Student Mistakes to Avoid

1. Misidentifying the Hypotenuse in Right Triangles

• Mistake: Writing $PT^2 = OP^2 + OT^2$ when using Pythagoras theorem on tangent triangle $\Delta OPT$.
• Correction: Always remember that the right angle is at the point of contact ($\angle OTP = 90^\circ$). Therefore, the hypotenuse is always the line segment joining the center to the external point ($OP$). The correct equation is: $OP^2 = OT^2 + PT^2$

2. Omitting Explicit Geometric References in Proofs

• Mistake: Stating $PA = PB$ or $OP \perp PT$ without giving reason statements in board exams.
• Correction: Loss of marks in CBSE board exams frequently occurs due to missing statements. Always write full justifications in parentheses:
  • Write: (Lengths of tangents drawn from an external point to a circle are equal) or (Theorem 10.2).
  • Write: (Radius is perpendicular to the tangent at the point of contact) or (Theorem 10.1).

3. Confusing Chord Bisectors with Tangent Properties

• Mistake: Assuming that any line from the center bisects a tangent line segment.
• Correction: A center line bisects a chord if perpendicular to it. For a tangent, the line from the center meets it at exactly one point (point of contact) perpendicular to it. The center line only bisects the angle between two tangents, not the tangent lines themselves.

4. Wrong Equations in Circumscribing Quadrilaterals

• Mistake: Equating adjacent sides instead of adding opposite sides when solving quadrilateral problems.
• Correction: Memorize the structural form: Sum of Opposite Sides is Equal ($AB + CD = AD + BC$), NOT $AB + BC = CD + DA$.

5. Practice Questions for Self-Assessment

Question 1

Problem: From a point $Q$, the length of the tangent to a circle is $24\text{ cm}$ and the distance of $Q$ from the center is $25\text{ cm}$. If $PA$ and $PB$ are two tangents to a circle with center $O$ such that $\angle APB = 80^\circ$, then find the value of $\angle POA$.

Solution Walkthrough:
1. Since PA and PB are tangents from P, angle APB = 80°.
2. Line OP bisects angle APB:
   Angle APO = 80° / 2 = 40°.
3. Radius OA is perpendicular to tangent PA, so angle OAP = 90°.
4. In right triangle OAP:
   Angle POA = 180° - (90° + 40°) = 180° - 130° = 50°.

Answer: $\angle POA = 50^\circ$.

Question 2

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

Solution Walkthrough:
1. Let AB be a diameter of a circle with center O.
2. Let line 'l' be the tangent at point A, and line 'm' be the tangent at point B.
3. By Theorem 10.1:
   Radius OA ⊥ line l  =>  Angle OAL = 90°
   Radius OB ⊥ line m  =>  Angle OBM = 90°
4. Consider line AB as a transversal intersecting lines l and m:
   Angle OAL and Angle OBM are alternate interior angles.
5. Since Angle OAL = Angle OBM = 90°, the alternate interior angles are equal.
6. Therefore, line l || line m (tangents are parallel).

Hence Proved.

Question 3

Problem: A triangle $ABC$ is drawn to circumscribe a circle of radius $4\text{ cm}$ such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of lengths $8\text{ cm}$ and $6\text{ cm}$ respectively. Find the sides $AB$ and $AC$.

Solution Walkthrough:
1. Let points of contact on AB and AC be E and F respectively.
2. By Theorem 10.2:
   BD = BE = 8 cm
   CD = CF = 6 cm
   AE = AF = x cm
3. Sides of ΔABC are:
   a = BC = 6 + 8 = 14 cm
   b = AC = (x + 6) cm
   c = AB = (x + 8) cm
4. Semi-perimeter s = (14 + x + 6 + x + 8) / 2 = (28 + 2x) / 2 = (14 + x) cm.
5. Area of ΔABC using Heron's Formula:
   Area = √[s(s - a)(s - b)(s - c)]
   Area = √[(14 + x)(14 + x - 14)(14 + x - x - 6)(14 + x - x - 8)]
   Area = √[(14 + x) · x · 8 · 6] = √[48x(14 + x)]
6. Area of ΔABC using sum of areas of ΔOBC, ΔOCA, ΔOAB with height r = 4 cm:
   Area = (1/2 · BC · r) + (1/2 · AC · r) + (1/2 · AB · r)
   Area = 1/2 · r · (a + b + c) = 1/2 · 4 · (28 + 2x) = 2(28 + 2x) = 4(14 + x)
7. Equate the two area expressions:
   √[48x(14 + x)] = 4(14 + x)
   Square both sides:
   48x(14 + x) = 16(14 + x)²
   Divide both sides by 16(14 + x):
   3x = 14 + x  =>  2x = 14  =>  x = 7 cm.
8. Therefore:
   AB = x + 8 = 7 + 8 = 15 cm
   AC = x + 6 = 7 + 6 = 13 cm

Answer: $AB = 15\text{ cm}$ and $AC = 13\text{ cm}$.

Question 4

Problem: Prove that the parallelogram circumscribing a circle is a rhombus.

Solution Walkthrough:
1. Let ABCD be a parallelogram circumscribing a circle.
2. Since ABCD is a circumscribing quadrilateral, by Example 3:
   AB + CD = AD + BC  --- (Equation 1)
3. Since ABCD is a parallelogram, opposite sides are equal:
   AB = CD  and  AD = BC
4. Substitute CD = AB and BC = AD into Equation 1:
   AB + AB = AD + AD
   2·AB = 2·AD  =>  AB = AD
5. Since adjacent sides AB and AD are equal, and opposite sides are equal, all four sides are equal:
   AB = BC = CD = DA
6. A parallelogram with all equal sides is a rhombus.

Hence Proved.

6. Exam Revision & FAQs

Q1: How many tangents can be drawn to a circle from a point inside, on, and outside the circle?

• Inside the circle: 0 tangents.
• On the circle: 1 tangent.
• Outside the circle: 2 tangents.

Q2: What is the relation between the angle between two tangents and the central angle subtended by their points of contact?

The angle between two tangents from an external point ($\angle APB$) and the angle subtended by the radii at the center ($\angle AOB$) are supplementary, meaning: $\angle APB + \angle AOB = 180^\circ$

Q3: What criteria of triangle congruence is strictly used to prove Theorem 10.2?

Theorem 10.2 uses the RHS (Right Angle-Hypotenuse-Side) congruence criterion. The right angle is formed by Theorem 10.1 ($\angle OAP = \angle OBP = 90^\circ$), the hypotenuse is the common segment $OP$, and one side is equal radii ($OA = OB$).

Q4: If two circles touch each other externally, how many common tangents can be drawn?

When two circles touch externally at a single point, exactly 3 common tangents can be drawn (two direct outer common tangents and one transverse common tangent passing through their point of contact).

                 Common Tangent 1
               ---------------------
                  .---.     .---.
                /       \ /       \
               |   O1   |X|   O2   |  <--- Common Tangent 3 (Transverse)
                \       / \       /
                  '---'     '---'
               ---------------------
                 Common Tangent 2