Circles
Introduction to Circles
The chapter Circles is an important geometry chapter in Class 10 Mathematics. In earlier classes, students learned basic concepts of circles such as radius, diameter, chord, and circumference. However, in Class 10, the focus is mainly on tangents to a circle and their properties.
A circle is one of the most common shapes we see in daily life. Examples include:
• Wheels of vehicles
• Coins
• Clocks
• Plates
• Rings
• Circular tracks
In mathematics, circles are used in geometry, engineering, architecture, astronomy, and many scientific fields.
This chapter mainly teaches:
• Tangent to a circle
• Properties of tangents
• Number of tangents from a point
• Equal tangents theorem
Understanding these concepts properly is important because theorem-based questions are frequently asked in board exams.
What is a Circle?
A circle is a collection of all points in a plane that are at the same distance from a fixed point.
The fixed point is called the centre of the circle.
For example:
If point O is the center of a circle and every point on the boundary is equally distant from O, then the figure formed is called a circle.
Important Parts of a Circle
Before studying tangents, students should know the basic terms.
Centre
The fixed point inside the circle is called the centre.
Example: Point O
Radius
The line segment joining the center to any point on the circle is called the radius.
Example: OA
All radii of a circle are equal.
Diameter
The line segment passing through the center and joining two points on the circle is called the diameter.
Relationship:
Diameter = 2 × Radius
Chord
A line segment joining any two points on a circle is called a chord.
The diameter is the longest chord of a circle.
Circumference
The boundary of a circle is called the circumference.
In simple words, it is the outer curved edge of the circle.
What is a Tangent to a Circle?
A tangent is a straight line that touches the circle at only one point.
That touching point is called the point of contact.
For example:
If line AB touches the circle at point P only, then AB is called a tangent.
Unlike a secant, a tangent does not cut through the circle.
Real-Life Examples of Tangents
Tangents can be seen in everyday life.
Examples include:
• A tyre touching the road
• A ladder touching a curved wall
• A road touching a circular park boundary
These examples help understand tangents better.
Important Theorem of Tangent
Theorem
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
This means:
If a tangent touches a circle at point P and O is the center, then the radius OP is perpendicular to the tangent PT.
Where:
OP = radius
PT = tangent
The angle formed is:
90°
This theorem is very important for board exams.
Example
Suppose:
Radius = 5 cm
A tangent touches the circle at point P.
Then:
The angle between radius and tangent will always be:
90°
Students should remember this theorem carefully.
Number of Tangents from a Point
The number of tangents depends on the position of the point.
Point Inside the Circle
If a point lies inside the circle, no tangent can be drawn.
Number of tangents = 0
Point On the Circle
If a point lies on the circle, only one tangent can be drawn.
Number of tangents = 1
Point Outside the Circle
If a point lies outside the circle, two tangents can be drawn.
Number of tangents = 2
This concept is commonly asked in theory questions.
Equal Tangents Theorem
Statement
The lengths of tangents drawn from an external point to a circle are equal.
Suppose point P lies outside the circle.
Two tangents:
PA and PB
touch the circle.
Then:
PA = PB
This theorem is extremely important for proofs and numerical questions.
Example
Suppose:
PA = 8 cm
Then:
PB = 8 cm
because tangents from the same external point are equal.
Proof of Equal Tangent Theorem (Basic Idea)
To prove:
PA = PB
We join:
OA and OB
which are radii to the points of contact.
We know:
• Radius is perpendicular to tangent
• OA = OB (radii of the same circle)
By using congruent triangles, we prove:
PA = PB
Students do not always need full proofs but should understand the concept.
Applications of Circles in Real Life
Wheels and Vehicles
Vehicle wheels are circular for smooth movement.
Architecture
Circular designs are used in buildings and monuments.
Sports
Footballs, basketballs, and circular tracks involve circles.
Engineering
Machines often use circular gears and rotating systems.
Astronomy
Scientists study circular orbits of planets.
Thus, circles are important in science and technology.
Solving Circle-Based Questions
Step 1: Draw the Figure
Always draw the circle properly.
Step 2: Mark Radius and Tangent
Identify:
• Radius
• Tangent
• Point of Contact
Step 3: Use Theorem
Remember:
Radius ⟂ Tangent
or
Tangents from the same point are equal
Step 4: Solve Carefully
Apply geometry rules properly.
This method helps avoid mistakes.
Common Mistakes Students Make
1. Confusing Chord and Tangent
Remember:
• Chord joins two points of a circle
• Tangent touches at one point only
2. Forgetting the 90° Rule
Radius and tangent always form:
90°
3. Wrong Tangent Lengths
Tangents from the same external point are always equal.
4. Poor Diagram
Many mistakes happen due to incorrect figures.
Always draw neat diagrams.
Tips to Score Good Marks in Circles
Learn all theorems properly.
Practice theorem-based proofs.
Understand tangent properties carefully.
Solve NCERT questions multiple times.
Revise diagrams regularly.
This chapter is usually easy and scoring in board exams.
Conclusion
The chapter Circles teaches important concepts related to tangents and their properties. Students learn how tangents behave, the relationship between tangents and radii, and the equality of tangents from an external point.
Key ideas such as radius perpendicular to tangent, equal tangent theorem, and number of tangents from a point are frequently asked in examinations.
Although the chapter may look theoretical at first, understanding the theorems and practicing diagrams makes it simple. With regular practice, students can easily score excellent marks in this chapter of Class 10 Mathematics.