Class 10 Maths Chapter 10 – Circles | Tangents, Theorems, NCERT Solutions & CBSE Questions (2025–26)

 Class 10 CBSE Maths – Chapter 10: Circles

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🔹 Key Concepts and Theorems

✅ Circle

• A circle is the set of all points in a plane that are equidistant from a fixed point (the centre).

• The fixed distance is called the radius.

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✅ Basic Terms

• Chord: A line segment joining two points on a circle.

• Diameter: A chord passing through the centre (longest chord).

• Tangent: A line that touches the circle at exactly one point.

• Point of Contact: The point where the tangent touches the circle.

• Secant: A line that intersects the circle at two distinct points.

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📘 Important Theorems

🔸 Theorem 1:

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

If a line touches a circle at only one point, then:
$\text{Radius} \perp \text{Tangent at the point of contact}$

🔸 Theorem 2:

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

If PA and PB are tangents from a point P to the circle at points A and B, then:
$PA = PB$

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📄 Exercise 10.1 – Conceptual (No written solution required)

Questions focus on conceptual understanding of:

• Number of tangents from different positions

• Defining tangents, chords, and radius

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📄 Exercise 10.2 – Tangent Problems

Q1. From a point Q outside a circle with centre O, the length of the tangent to the circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius.

$OQ^2 = OQ^2 = r^2 + 24^2 = 25^2 \Rightarrow r^2 = 625 - 576 = 49 \Rightarrow r = 7 \text{ cm}$

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Q2. A point P is 29 cm from the centre of a circle. The length of the tangent from P to the circle is 20 cm. Find the radius of the circle.

$OP^2 = r^2 + 20^2 = 29^2 \Rightarrow r^2 = 841 - 400 = 441 \Rightarrow r = 21 \text{ cm}$

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Q3. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that $\angle PTQ = 2\angle OPQ$

Solution Outline:

• Use the fact that $\angle OTP = \angle OTQ = 90^\circ$

• Triangles OTP and OTQ are congruent

• Use geometry and triangle properties to prove the relationship

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🔁 Summary and Revision Notes

• A tangent touches the circle at exactly one point.

• Tangent is always perpendicular to the radius at the point of contact.

• From a point outside the circle, two tangents of equal length can be drawn.

• Right triangle is formed with radius, tangent, and line from external point to centre.

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📘 CBSE Previous Year Questions – Circles

✅ 1-Mark

(2020) Define tangent. How many tangents can be drawn from a point inside the circle?
Answer: A line touching the circle at one point is called a tangent. From a point inside the circle, no tangent can be drawn.

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✅ 2-Mark

(2022) From a point 13 cm away from the centre of a circle, a tangent of length 5 cm is drawn. Find the radius of the circle.
$r^2 = 13^2 - 5^2 = 169 - 25 = 144 \Rightarrow r = 12 \text{ cm}$

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✅ 3-Mark

(2018) Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that $\angle OTQ = \angle OTP$
Proof Sketch:

• Tangents from T are equal → TP = TQ

• $OT = OT$ (common side)

• Triangles OTP and OTQ are congruent by RHS

• So, $\angle OTQ = \angle OTP$