Circles
In Class 10, the chapter on Circles focuses on tangents — straight lines that touch a circle at exactly one point. You will learn how many tangents can be drawn from a point, the key fact that a tangent is perpendicular to the radius at the point of contact, and that the two tangents drawn from an external point are equal in length. These two theorems drive almost every question.
Learning objectives
- Define a tangent and identify the point of contact.
- State how many tangents pass through points inside, on, and outside a circle.
- Use the fact that the tangent is perpendicular to the radius at the contact point.
- Use the theorem that tangents from an external point are equal.
- Solve numerical problems on tangent lengths.
Key concepts
Tangent to a circle
A tangent is a line that touches a circle at exactly one point, called the point of contact. A secant, by contrast, cuts the circle at two points.
Number of tangents from a point
From a point inside the circle, no tangent can be drawn. From a point on the circle, exactly one tangent can be drawn. From a point outside the circle, exactly two tangents can be drawn.
Tangent perpendicular to radius
The tangent at any point of a circle is perpendicular to the radius drawn to that point of contact. This gives a right angle that is used in most tangent calculations.
Equal tangents from an external point
The lengths of the two tangents drawn from an external point to a circle are equal. Combined with the right angle above, the tangent length from a point at distance d from the centre is √(d² − r²).
Important formulas
Tangent length
length = √(d² − r²)
d = distance of the external point from the centre, r = radius.
Angle with radius
tangent ⟂ radius at the point of contact (90°)
Key definitions
- Tangent
- A line touching a circle at exactly one point.
- Point of contact
- The single point where a tangent touches the circle.
- Secant
- A line that intersects a circle at two points.
Solved examples
Q1. From a point 13 cm from the centre of a circle of radius 5 cm, find the length of the tangent.
Solution: length = √(d² − r²) = √(13² − 5²) = √(169 − 25) = √144 = 12 cm.
Q2. The tangent from a point at distance 10 cm from the centre is 8 cm long. Find the radius.
Solution: length = √(d² − r²) ⇒ 8 = √(100 − r²) ⇒ 64 = 100 − r² ⇒ r² = 36 ⇒ r = 6 cm.
Q3. Two tangents PA and PB are drawn from an external point P. If PA = 7 cm, what is PB?
Solution: Tangents from an external point are equal, so PB = PA = 7 cm.
Common mistakes to avoid
- Thinking a tangent meets the circle at two points (that is a secant).
- Forgetting the right angle between the tangent and the radius.
- Using d² + r² instead of d² − r² for the tangent length.
- Assuming a point inside the circle has tangents (it has none).
Circles — MCQ Quiz
12 questions with instant feedback. Use number keys 1–4 to answer.
A tangent to a circle touches it at:
Practice questions
Short answer
How many tangents can a circle have in all?
Infinitely many — one at every point of the circle.
State the angle between a tangent and the radius at the point of contact.
90° (they are perpendicular).
If one tangent from an external point is 9 cm, what is the other?
9 cm — tangents from an external point are equal.
Long answer
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact (state the idea).
Take the centre O and tangent at point P. Any point Q on the tangent other than P lies outside the circle, so OQ > OP. Hence OP is the shortest distance from O to the tangent, and the shortest distance is the perpendicular — so OP ⟂ the tangent.
Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle that is tangent to the smaller one.
The tangent touches the smaller circle, so the radius (3 cm) is perpendicular to the chord and bisects it. Half-chord = √(5² − 3²) = √16 = 4 cm, so the full chord = 8 cm.
HOTS (Higher Order Thinking)
A quadrilateral ABCD is drawn so that all four sides touch a circle. Show that AB + CD = AD + BC.
Using equal tangents from each vertex, the tangent lengths satisfy AB + CD = (AP + PB) + (CR + RD) and AD + BC = (AS + SD) + (BQ + QC); equal tangents pair up to make both sums equal.
Why can't a tangent be drawn from a point inside the circle?
Any line through an interior point must cross the circle at two points (a secant), so it can never touch at just one point.
Quick revision
Revision notes
- Tangent: touches the circle at exactly one point.
- Tangents from a point: inside 0, on 1, outside 2.
- Tangent ⟂ radius at the point of contact (90°).
- Tangents from an external point are equal; length = √(d² − r²).
Key takeaways
- The right angle between tangent and radius unlocks most numericals.
- Equal tangents from a point simplify perimeter and quadrilateral problems.
- Distinguish tangent (one point) from secant (two points).
Frequently asked questions
What is the main idea of the Class 10 Circles chapter?
Tangents — how many can be drawn from a point, that a tangent is perpendicular to the radius, and that tangents from an external point are equal.
How do I find the length of a tangent?
Use √(d² − r²), where d is the distance of the point from the centre and r is the radius.
Is a tangent the same as a chord?
No. A tangent touches the circle at one point; a chord joins two points on the circle.