Circles — Class 10 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Circles" — 8 important questions with detailed answers for CBSE board exam preparation.

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TL;DR: Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Circles" — 8 important questions with detailed answers for CBSE board exam prepara…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

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Key Questions Covered:

State and prove the theorem: A tangent to a circle is perpendicular to the ra…
Two tangents are drawn to a circle from an external point. Prove that they ar…
A circle with center O and radius 5 cm has a point P at distance 13 cm from t…
A chord of a circle is 24 cm long. The radius of the circle is 13 cm. Find th…
Two circles with centers O₁ and O₂ and radii 8 cm and 5 cm respectively touch…
A circle has a radius of 10 cm. A point P is at distance 26 cm from the cente…
+ 2 more questions in the full chapter

Solutions Summary:

Question	Status
State and prove the theorem: A tangent to a circle is per…	✓ Solved
Two tangents are drawn to a circle from an external point…	✓ Solved
A circle with center O and radius 5 cm has a point P at d…	✓ Solved
A chord of a circle is 24 cm long. The radius of the circ…	✓ Solved
Two circles with centers O₁ and O₂ and radii 8 cm and 5 c…	✓ Solved
A circle has a radius of 10 cm. A point P is at distance …	✓ Solved

Showing 6 of 8 questions

Q1: State and prove the theorem: A tangent to a circle is perpendicular to the radius at the point of tangency.

Theorem: A tangent to a circle is perpendicular to the radius at the point of tangency. Given: Let O be the center of a circle. Let AB be a tangent to the circle at point P. OP is the radius to point P. Prove: OP ⊥ AB (that is, OP is perpendicular to AB) Proof: Step 1: Assume, for contradiction, that OP is not perpendicular to AB. Suppose OP makes an angle other than 90° with the tangent AB. Step 2: From O, draw a perpendicular to the tangent AB, meeting it at point Q. Since this perpendicu...

Q2: Two tangents are drawn to a circle from an external point. Prove that they are equal in length.

Theorem: Two tangents drawn to a circle from an external point are equal in length. Given: Let O be the center of the circle. Let P be an external point. PA and PB are two tangents to the circle at points A and B respectively. Prove: PA = PB Proof: Step 1: Since PA and PB are tangents, we use the perpendicularity property. OA ⊥ PA (radius is perpendicular to tangent at point of contact) OB ⊥ PB (radius is perpendicular to tangent at point of contact) Step 2: Consider triangles OAP and OBP. ...

Q3: A circle with center O and radius 5 cm has a point P at distance 13 cm from the center. From P, two tangents are drawn to the circle. Find the length of each tangent.

Given: Circle with center O and radius r = 5 cm Point P at distance OP = 13 cm from center Two tangents PA and PB are drawn from P to the circle Find: Length of each tangent (PA and PB) Diagram (text description): A (point of tangency) /| / | / | 5 cm (radius OA) / | / | / O________ / 13 cm / P \_____ B (point of tangency) Solution: Step 1: Since PA is a tangent at point A: OA ⊥ PA Therefore, triangle OAP ...

Q4: A chord of a circle is 24 cm long. The radius of the circle is 13 cm. Find the distance of the chord from the center of the circle.

Given: Length of chord = 24 cm Radius of circle = 13 cm Find: Distance of chord from the center Diagram (text description): O (center) | | d (perpendicular distance) | ----M---- (M is foot of perpendicular, midpoint of chord) / | \ A | B |--12-|-12--| (half-chord on each side) \__________/ Chord AB = 24 cm Key Property: The perpendicular from the center of a circle to a chord bisects the chord. Step 1: Let O be the center and AB ...

Q5: Two circles with centers O₁ and O₂ and radii 8 cm and 5 cm respectively touch each other externally. Find the distance between their centers.

Given: Circle 1: Center O₁, radius r₁ = 8 cm Circle 2: Center O₂, radius r₂ = 5 cm The circles touch each other externally Find: Distance between centers O₁O₂ Diagram (text description): Circle 1 Point of tangency Circle 2 * * * \ / \ / \ 8 cm / \ 5 cm / \ O₁ O₂ / *-----------•---------•-----------* 8 cm ? 5 cm Definition: Two c...

Q6: A circle has a radius of 10 cm. A point P is at distance 26 cm from the center. From P, a secant is drawn that passes through the center O. This secant intersects the circle at points A and B (where A is closer to P). Find PA and PB.

Given: Circle with center O and radius r = 10 cm Point P at distance OP = 26 cm from center Secant through P and O intersects circle at A and B A is closer to P than B Find: PA and PB Diagram (text description): A O B •----•---------•----P 10 cm 10 cm 16 cm Secant line through O Solution: Step 1: Understand the configuration. Since the secant passes through the center O, it's actually a diameter extended to point P. Let's place O at ...

Showing 6 of 8 questions. Visit the full page for complete solutions.