Chapter 10 · Class 10 CBSE · MCQ Test
Circles MCQ Test — Class 10 CBSE
Practice 10 multiple-choice questions with instant answer reveal and explanations.
Circles — MCQ Questions
1Which of the following statements correctly defines a tangent to a circle?
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Answer: A line that touches a circle at exactly one point.
Hint: Recall the fundamental difference between a tangent, a secant, and a chord.
Solution:
A tangent is defined as a line that intersects the circle at exactly one point. This point is called the point of contact.
Options A describes a secant, option B describes a line containing a diameter, and option D describes a chord.
2From a point P lying *inside* a circle, how many tangents can be drawn to the circle?
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Answer: Zero
Hint: Consider if a line starting from inside the circle can touch it at only one point without passing through.
Solution:
If a point P lies inside a circle, any line passing through P will necessarily intersect the circle at two distinct points.
By the definition of a tangent (touching at exactly one point), no tangent can be drawn to the circle from a point lying inside it.
3A tangent PQ at a point P of a circle of radius 5 cm meets a line through the center O at a point Q so that OQ = 12 cm. What is the measure of ∠OPQ?
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Answer: 90°
Hint: Recall the relationship between the radius and the tangent at the point of contact.
Solution:
According to Theorem 1, the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Therefore, the radius OP is perpendicular to the tangent PQ at point P.
This means ∠OPQ = 90°.
4A student states that if a line AB is tangent to a circle at point P, and OP is a radius, then ∠OPA must be 60°. What is the error in this statement?
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Answer: The angle should be 90°.
Hint: Remember the specific angle formed between a radius and a tangent at the point of contact.
Solution:
Theorem 1 states that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Perpendicular lines form an angle of 90°.
Therefore, ∠OPA must be 90°, not 60°.
5Which of the following statements about a tangent to a circle is TRUE?
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Answer: The radius drawn to the point of contact is perpendicular to the tangent.
Hint: Focus on the relationship between the radius and the tangent *at the point of contact*.
Solution:
Option A is false; a tangent does not pass through the center. Option C is false; a tangent intersects at exactly one point. Option D is false; there is no such general rule for the length of a tangent.
Theorem 1 states that the radius drawn to the point of contact is perpendicular to the tangent. This statement is correct.
6From an external point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the center is 25 cm. The radius of the circle is:
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Answer: 7 cm
Hint: Draw a diagram. The radius, the tangent, and the line connecting the external point to the center form a right-angled triangle.
Solution:
Let O be the center of the circle and P be the point of contact of the tangent from Q. According to Theorem 1, OP ⊥ PQ, forming a right-angled triangle ΔOPQ at P.
We are given PQ = 24 cm (tangent length) and OQ = 25 cm (hypotenuse). Let OP = r (radius).
Using the Pythagorean theorem: OP² + PQ² = OQ²
r² + 24² = 25² ⇒ r² + 576 = 625 ⇒ r² = 625 - 576 ⇒ r² = 49 ⇒ r = 7 cm.
7Two tangents PA and PB are drawn from an external point P to a circle with center O. Which of the following is the correct reason why PA = PB?
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Answer: Triangles ΔPAO and ΔPBO are congruent.
Hint: Think about proving the equality of line segments in geometry. What method is typically used?
Solution:
To prove PA = PB, we establish that the triangles formed (ΔPAO and ΔPBO) are congruent.
We know OA = OB (radii), OP = OP (common side), and ∠OAP = ∠OBP = 90° (radius ⊥ tangent).
By RHS (Right angle - Hypotenuse - Side) congruence criterion, ΔPAO ≅ ΔPBO. Therefore, by CPCTC, PA = PB.
8Consider a circle with center O and a point A on the circle. A line segment OA is drawn. Another line M touches the circle at point A and extends infinitely in both directions. Which statement is correct?
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Answer: OA is a radius and M is a tangent.
Hint: Recall the precise definitions of radius and tangent.
Solution:
A line segment connecting the center of a circle (O) to a point on the circle (A) is defined as a radius. So, OA is a radius.
A line that touches a circle at exactly one point (A) and extends infinitely in both directions is defined as a tangent. So, M is a tangent.
9A line is drawn through the end-point of a radius of a circle and is perpendicular to it. What can be concluded about this line?
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Answer: It must be a tangent.
Hint: This question tests the converse of the theorem about the relationship between a radius and a tangent.
Solution:
Theorem 1 states that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
The converse of this theorem states that if a line drawn through the end-point of a radius is perpendicular to the radius, then it must be a tangent to the circle at that end-point.
10In a circle with center O, tangents PT and PU are drawn from an external point P. If PT = 8 cm and the perimeter of ΔPTO is 20 cm, what is the length of PU?
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Answer: 8 cm
Hint: Recall the property of tangents drawn from an external point to a circle.
Solution:
According to Theorem 2, the lengths of tangents drawn from an external point to a circle are equal.
Given that PT and PU are tangents drawn from the external point P, it must be that PT = PU.
Since PT = 8 cm, then PU must also be 8 cm.
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Tips for Circles MCQs
- 1Read each question carefully and identify what is being asked before looking at the options.
- 2Try to solve the problem mentally or on paper first, then match your answer with the options.
- 3Use elimination — rule out clearly wrong options to improve your chances even when unsure.
- 4Check units, signs, and edge cases — these are common traps in Circles MCQs.
- 5Review your mistakes after completing the test to build lasting understanding.
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