Class 10 Mathematics
Chapter 10: Circles
Exercise 10.1 – Stepwise Solutions
1. How many tangents can a circle have?
A tangent can be drawn at every point of the circle.
Since a circle has infinitely many points,
A circle can have infinitely many tangents.
2. Fill in the blanks:
(i) A tangent to a circle intersects it in ______ point(s).
A tangent touches the circle at only one point.
Answer: One
(ii) A line intersecting a circle in two points is called a ______.
A line that cuts a circle at two distinct points is called a secant.
Answer: Secant
(iii) A circle can have ______ parallel tangents at the most.
At most, two tangents can be parallel to each other (on opposite sides of the circle).
Answer: Two
(iv) The common point of a tangent to a circle and the circle is called ______.
The point where the tangent touches the circle is called the point of contact.
Answer: Point of contact
3. MCQ Problem
Given:
Radius = 5 cm
OQ = 12 cm
PQ is tangent at P
Important Property:
The radius drawn to the point of contact is perpendicular to the tangent.
So, ∠OPQ = 90°
Triangle OPQ is right-angled.
Apply Pythagoras Theorem:
OQ² = OP² + PQ²
12² = 5² + PQ²
144 = 25 + PQ²
PQ² = 144 − 25
PQ² = 119
PQ = √119 cm
Correct Option: (D) √119 cm
4. Construction Question
Steps to draw:
- Draw a given line l.
- Draw a circle with centre O and suitable radius.
- Draw a line parallel to l that just touches the circle at one point (this will be the tangent).
- Draw another line parallel to l that cuts the circle at two points (this will be the secant).
Thus, one line is tangent (touches at one point) and the other is secant (intersects at two points).
Final Answers Summary
- 1) Infinitely many tangents
- 2(i) One
- 2(ii) Secant
- 2(iii) Two
- 2(iv) Point of contact
- 3) √119 cm