CBSE Class 10 Mathematics (2026–27)
Chapter 10: Circles
20 Important Questions and Answers
The chapter mainly focuses on tangents to a circle and their properties. The two key theorems are: (i) the tangent at any point of a circle is perpendicular to the radius through the point of contact, and (ii) tangents drawn from an external point to a circle are equal in length.
Q1. What is a tangent to a circle? How is it different from a secant?
Answer:
A tangent is a straight line that touches a circle at exactly one point, known as the point of contact. At this point, the tangent does not cut the circle. A secant, on the other hand, intersects the circle at two distinct points. A tangent can be considered a special case of a secant when the two points of intersection merge into one. Tangents are important because they help establish several geometric properties of circles. The point where the tangent touches the circle is unique, and only one tangent can be drawn at a given point on the circumference.
Q2. State and explain the theorem related to a tangent and radius.
Answer:
The theorem states that the tangent at any point of a circle is perpendicular to the radius passing through the point of contact. Suppose a tangent touches the circle at point P and O is the centre. Then OP is the radius and the angle between OP and the tangent is 90°. This property helps in solving many geometrical problems involving circles. The theorem is based on the fact that the radius is the shortest distance from the centre to the tangent line. Therefore, the radius must meet the tangent at a right angle.
Q3. Why can only one tangent be drawn at a point on a circle?
Answer:
At any point on a circle, only one radius can be drawn from the centre to that point. According to the tangent-radius theorem, the tangent must be perpendicular to this radius. Since only one line can be drawn perpendicular to a given line at a specific point, only one tangent is possible. If more than one tangent were drawn, the radius would have more than one perpendicular at the same point, which is impossible in geometry. Hence, exactly one tangent can be drawn at any point on the circle.
Q4. How many tangents can be drawn from a point inside, on, and outside a circle?
Answer:
The number of tangents depends on the position of the point relative to the circle. From a point inside the circle, no tangent can be drawn because every line through the point intersects the circle. From a point on the circle, exactly one tangent can be drawn. From a point outside the circle, exactly two tangents can be drawn touching the circle at different points. These tangents have equal lengths. Understanding these cases helps in solving construction and proof-based questions related to circles and tangents.
Q5. State the theorem of equal tangents.
Answer:
The theorem states that the lengths of tangents drawn from an external point to a circle are equal. If PA and PB are tangents drawn from an external point P to a circle touching it at A and B respectively, then PA = PB. This theorem is proved using congruent triangles formed by joining the centre to the points of contact. Since the radii are equal and the tangents are perpendicular to the radii, the RHS congruence criterion applies. This theorem is frequently used in board examination questions.
Q6. A tangent touches a circle of radius 5 cm. If the distance from the centre to an external point is 13 cm, find the tangent length.
Answer:
Let O be the centre and P the external point. Radius OA = 5 cm and OP = 13 cm. Since OA is perpendicular to the tangent AP, triangle OAP is right-angled.
Using Pythagoras theorem:
AP² = OP² – OA²
AP² = 13² – 5²
AP² = 169 – 25
AP² = 144
AP = 12 cm
Therefore, the length of the tangent is 12 cm. This problem demonstrates the use of the tangent-radius theorem and right-angled triangles in circles.
Q7. Why is the radius called the normal to the tangent?
Answer:
A normal is a line that is perpendicular to another line at the point of contact. In a circle, the radius drawn to the point where the tangent touches the circle forms a right angle with the tangent. Therefore, the radius acts as a normal to the tangent. This property is fundamental in circle geometry and helps establish relationships between tangents, radii and angles. Whenever a tangent is involved, the radius to the point of contact can immediately be considered perpendicular to it.
Q8. Can a circle have more than two parallel tangents? Explain.
Answer:
No, a circle can have at most two parallel tangents. These tangents touch the circle at opposite ends of a diameter. The distance between the two tangents equals the diameter of the circle. Any third line parallel to these tangents would either not touch the circle at all or would cut the circle at two points, becoming a secant. Therefore, only two parallel tangents are possible for a given circle. This property is useful in understanding the geometric arrangement of tangents around a circle.
Q9. Define the point of contact.
Answer:
The point of contact is the single point where a tangent touches a circle. It is the only common point between the tangent and the circle. At this point, the radius drawn from the centre is perpendicular to the tangent. The point of contact plays an important role in proving theorems related to tangents and circles. In geometric constructions, identifying the point of contact correctly helps determine right angles and equal tangent lengths from external points.
Q10. What is the relationship between tangent and radius at the point of contact?
Answer:
The tangent and radius are always perpendicular at the point of contact. If a tangent touches the circle at point P and O is the centre, then OP is the radius and ∠OPT = 90°. This relationship is one of the most important properties of circles. It is frequently used to prove congruence of triangles, calculate lengths, and determine unknown angles in geometrical figures involving circles and tangents.
Q11. Why are tangents from an external point equal?
Answer:
Let PA and PB be tangents from an external point P. Join OA, OB and OP where O is the centre. OA and OB are radii, so they are equal. OP is common to both triangles. Also, OA and OB are perpendicular to the tangents. Thus, triangles OAP and OBP are congruent by RHS criterion. Therefore, corresponding sides PA and PB are equal. This theorem is widely applied in solving geometric proofs and finding unknown lengths in circles.
Q12. If PA and PB are tangents from an external point P and PA = 8 cm, find PB.
Answer:
According to the theorem of equal tangents, tangents drawn from the same external point to a circle are equal in length. Therefore:
PA = PB
Given PA = 8 cm
Hence,
PB = 8 cm
This theorem simplifies many geometry problems because once one tangent length is known, the other tangent from the same external point automatically becomes equal. It is one of the most frequently used properties in Class 10 circle problems and proofs.
Q13. What is a secant?
Answer:
A secant is a line that intersects a circle at two distinct points. Unlike a tangent, which touches the circle at only one point, a secant passes through the circle. The portion of the secant inside the circle forms a chord. A tangent can be viewed as a limiting position of a secant when the two intersection points coincide. Understanding the difference between tangents and secants is important for identifying the correct theorem and solving geometry problems involving circles.
Q14. Explain why a tangent is a special case of a secant.
Answer:
A secant intersects a circle at two points. As these two points move closer together, the chord between them becomes smaller. When the two points coincide at a single point, the secant touches the circle at exactly one point. In this limiting position, the secant becomes a tangent. Therefore, a tangent is regarded as a special case of a secant. This concept helps students understand the geometric relationship between different lines associated with circles.
Q15. What is the maximum number of common points between a line and a circle?
Answer:
A line and a circle can have a maximum of two common points. If they have no common point, the line is non-intersecting. If they have one common point, the line is a tangent. If they have two common points, the line is a secant. Since a circle is a closed curve, a straight line cannot intersect it at more than two points. This classification forms the basis of understanding tangents and secants in geometry.
Q16. What is meant by an external point of a circle?
Answer:
An external point is a point lying outside the circle. The distance of this point from the centre is greater than the radius of the circle. From such a point, exactly two tangents can be drawn to the circle. These tangents touch the circle at two different points and are equal in length. External points are important in geometry because most tangent-related theorems and constructions involve tangents drawn from an external point.
Q17. If the radius of a circle is 7 cm, what is the distance between two parallel tangents?
Answer:
The distance between two parallel tangents to a circle is equal to the diameter of the circle. Since the radius is 7 cm:
Diameter = 2 × Radius
= 2 × 7
= 14 cm
Therefore, the distance between the two parallel tangents is 14 cm. This result follows from the fact that the tangents touch the circle at opposite ends of a diameter and remain parallel to each other.
Q18. Why is a tangent considered the shortest line from the circle to the tangent point?
Answer:
The radius drawn to the point of contact is perpendicular to the tangent. In geometry, the perpendicular distance from a point to a line is always the shortest distance. Therefore, the radius represents the minimum distance between the centre of the circle and the tangent. This property is used in proving that the tangent is perpendicular to the radius and helps establish many other circle theorems and constructions.
Q19. What are the two most important theorems of the chapter?
Answer:
The first theorem states that the tangent at any point of a circle is perpendicular to the radius through the point of contact. The second theorem states that the lengths of tangents drawn from an external point to a circle are equal. These two theorems form the foundation of the entire chapter. Most numerical and proof-based questions are solved using these results. A strong understanding of these theorems enables students to solve board examination questions efficiently and accurately.
Q20. Why is Chapter “Circles” important for CBSE Board Exams?
Answer:
The chapter “Circles” is important because it contains fundamental geometric theorems that are frequently tested in CBSE examinations. Questions may involve proving tangent properties, finding unknown lengths, solving right-triangle applications, and using equal tangent concepts. The chapter also develops logical reasoning and proof-writing skills. Since only a few core concepts are involved, students who thoroughly understand the theorems and practice NCERT questions can score well. Regular revision of tangent properties and solved examples is the key to mastering this chapter.