Class 10 Maths · Original EZT practice
Circles (Class 10 Chapter 10): summary and practice
Circles studies tangents: the tangent is perpendicular to the radius at the contact point, and the two tangents from one external point are equal - giving the tangent length sqrt(d^2 - r^2). All questions are original EZT tutor practice following the chapter's own section order.
6 original worked examples, 6 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 21 October 2026:
1. Secant and tangent
A secant cuts the circle twice; a tangent touches it at exactly one point - the limiting position of a secant as its two intersections merge.
2. Tangent perpendicular to radius
The tangent at any point is perpendicular to the radius through the contact point. This single theorem builds the right triangle behind every tangent-length question.
3. How many tangents from a point
Inside the circle: none. On the circle: exactly one. Outside: exactly two - one to each side.
4. Equal tangent lengths
The lengths of the two tangents from one external point are equal. With the perpendicularity theorem, that length is sqrt(d^2 - r^2), d being the distance from the centre.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Tangent length from an external point
Choose the method: Pythagoras on radius, tangent and centre-line: tangent = sqrt(d^2 - r^2).
EZT practice question
A point is 13 cm from the centre of a circle of radius 5 cm. Find the tangent length.
Show the worked solution
- The radius meets the tangent at 90 degrees.
- tangent^2 = 13^2 - 5^2 = 169 - 25 = 144.
- tangent = 12.
Answer: 12 cm.
Check: 5-12-13 is a Pythagorean triple - the right triangle closes exactly.
Watch for this: Subtract INSIDE the root: sqrt(169 - 25), not 13 - 5.
Now try it yourself
Radius 6 cm, point 10 cm from the centre - tangent?
Check your answer
8 cm.
sqrt(100 - 36) = sqrt(64).
Find the distance from a tangent length
Choose the method: Same triangle, other unknown: d = sqrt(r^2 + t^2).
EZT practice question
The tangent from a point to a circle of radius 3 cm is 4 cm long. How far is the point from the centre?
Show the worked solution
- d^2 = r^2 + t^2 = 9 + 16 = 25.
- d = 5.
- Done.
Answer: 5 cm.
Check: 3-4-5 triple again - the radius, tangent and centre-line always form a right triangle.
Watch for this: Here you ADD the squares - the centre-line is the hypotenuse, the tangent is a leg.
Now try it yourself
Radius 5 cm, tangent 12 cm - distance?
Check your answer
13 cm.
sqrt(25 + 144).
Use equal tangents from one point
Choose the method: TP = TQ whenever both touch the circle from the same external point T.
EZT practice question
Tangents TP and TQ touch a circle from point T. If TP = 7.5 cm, what is TQ?
Show the worked solution
- Both tangents start at the same external point T.
- The equal-tangents theorem applies.
- TQ = TP.
Answer: TQ = 7.5 cm.
Check: The proof uses two congruent right triangles sharing the hypotenuse TO - the lengths must match.
Watch for this: Equality holds only for tangents from the SAME point - tangents from different points can differ.
Now try it yourself
TP = 9 cm from T; a tangent from another point S is 12 cm. Is that possible?
Check your answer
Yes - S is a different external point, so its tangent length is its own.
Equality binds TP and TQ only.
Count the tangents from a position
Choose the method: Compare the point's distance from the centre with the radius.
EZT practice question
A circle has radius 4 cm. How many tangents can be drawn from points A (3 cm from centre), B (4 cm) and C (9 cm)?
Show the worked solution
- A: 3 < 4, inside the circle.
- B: 4 = 4, on the circle.
- C: 9 > 4, outside.
Answer: From A: none. From B: one. From C: two.
Check: Inside points cannot reach the boundary with a touching line; on the circle there is exactly the one at that point; outside always gives a symmetric pair.
Watch for this: Inside means ZERO tangents - not one.
Now try it yourself
How many tangents from a point 1 cm from the centre of a 4 cm circle?
Check your answer
None.
The point is inside.
Quadrilateral around a circle
Choose the method: Equal tangents at each corner force AB + CD = AD + BC.
EZT practice question
A quadrilateral ABCD circumscribes a circle. Show why AB + CD = AD + BC.
Show the worked solution
- From each vertex, the two tangents to the circle are equal.
- AB + CD collects all eight tangent pieces one way.
- AD + BC collects the same eight pieces the other way.
Answer: The sums of opposite sides are equal: AB + CD = AD + BC.
Check: For a rhombus (all sides s) the claim reads 2s = 2s - consistent, and every rhombus does circumscribe a circle.
Watch for this: It is OPPOSITE sides whose sums match - adjacent sums need not.
Now try it yourself
If AB = 8, CD = 5 and AD = 7, find BC.
Check your answer
BC = 6.
8 + 5 = 7 + BC.
Secant versus tangent
Choose the method: Count the intersection points: two means secant, one means tangent.
EZT practice question
A line meets a circle in exactly one point P. Is it a tangent, and what angle does it make with the radius OP?
Show the worked solution
- One intersection point: tangent by definition.
- The tangent-radius theorem applies at P.
- The angle at P is 90 degrees.
Answer: Yes, a tangent - and it is perpendicular to OP.
Check: Both facts come from the chapter's two opening ideas: the definition and Theorem 10.1.
Watch for this: A line touching the circle is not free to make any angle - the perpendicularity is a theorem, not a choice.
Now try it yourself
A line cuts the circle at two points. What is it called?
Check your answer
A secant.
Two intersections.
Try a mixed set
Choose a method or explanation before opening each answer.
1. Tangent meets radius at?
Show answer and reason
90 degrees.
Theorem 10.1.
2. Tangent length formula?
Show answer and reason
sqrt(d^2 - r^2).
Pythagoras with d as hypotenuse.
3. Tangents from an external point?
Show answer and reason
Two, and equal in length.
Theorem 10.2.
4. Opposite sides of a circumscribing quadrilateral?
Show answer and reason
AB + CD = AD + BC.
Equal tangents at each vertex.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Radius 6 cm, point 10 cm from the centre - tangent length.
- Radius 3 cm, tangent 4 cm - distance from the centre.
- Tangents from a point 2 cm from the centre of a 5 cm circle.
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Questions about this practice page
- How many tangents can be drawn to a circle from a point?
- It depends on where the point stands relative to the circle: none from inside, exactly one from a point on the circle (the tangent at that point), and exactly two from any external point.
- Why are the two tangent lengths from one external point equal?
- Join the external point to the centre. That line is the shared hypotenuse of two right triangles whose legs are the radius and each tangent. The triangles are congruent, so the tangents match.
- How do I find the tangent length without drawing anything?
- Use tangent = sqrt(d^2 - r^2), where d is the point's distance from the centre and r is the radius. It is the Pythagoras theorem on the right triangle formed by the radius, the tangent and the centre-line.
- What is the difference between a secant and a tangent?
- A secant crosses the circle at two points; a tangent touches it at exactly one. Picture moving a secant so its two crossing points slide together - when they merge, the line has become a tangent.
- Are these the NCERT exercise questions for Circles (Class 10)?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.