Class 9 NCERT Solutions - Chapter 10 Circles - Exercise 10.1

Chapter 10 of the Class 9 NCERT Mathematics book covers Circles focusing on the fundamental properties and theorems related to the circles. Exercise 10.1 is designed to help students understand and apply these properties through the various problems. This exercise is essential for grasping the basic concepts of the circles including their definitions, parts, and properties.

What is a Circle?

A circle is a two-dimensional geometric shape consisting of all points that are equidistant from the fixed point called the center. The distance between any point on the circle and the center is called the radius. The longest distance across the circle passing through the center is known as the diameter. Key components of a circle include:

• Center (O): The fixed point from which every point on the circle is equidistant.
• Radius (r): The distance from the center to any point on the circle.
• Diameter (d): Twice the radius of the longest chord of the circle.
• Chord: A line segment with both the endpoints on the circle.
• Arc: A part of the circumference of the circle.

Question 1:Fill in the blanks

(i) The centre of a circle lies in ____________ of the circle. (exterior/ interior)

Answer: Interior.

(ii) A point, whose distance from the centre of a circle is greater than its radius lies in ______________ of the circle. (exterior/ interior)

Answer: Exterior.

(iii) The longest chord of a circle is a ______________ of the circle.

Answer: Diameter.

(iv) An arc is a _____________ when its ends are the ends of a diameter.

Answer: Semicircle.

(v) Segment of a circle is the region between an arc and _____________ of the circle.

Answer: Chord.

(vi) A circle divides the plane, on which it lies, in _______________ parts.

Answer: Three.

Problem 2: Write True or False: Give reasons for your answers. 

(i) Line segment joining the centre to any point on the circle is a radius of the circle.

Answer: True, Any line segment drawn from the centre of the circle to any point on it is called radius of the circle and will be of equal length.

(ii) A circle has only finite number of equal chords.

Answer: False, There can be infinite number of equal chords in a circle.

(iii) If a circle is divided into three equal arcs, each is a major arc. 

Answer: False, When the arcs are not equal we will have major and minor arc, equal arcs cannot be classified as a major arc or a minor arc.

(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.

Answer: True, Diameter is the longest chord of the circle and the length of the longest chord in a circle is twice the length of radius of the circle.

(v) Sector is the region between the chord and its corresponding arc.

Answer: False, Sector is defined as the region between the arc and 2 radii of the circle.

(vi) A circle is a plane figure

Answer: True, A circle is a 2D figure which can be drawn on a plane.

Conclusion

Understanding circles and their properties is crucial for the solving various geometric problems. Exercise 10.1 in Chapter 10 of the Class 9 NCERT Mathematics book provides the foundational practice to help students become proficient in working with the circles. Mastery of these concepts will enhance problem-solving skills and prepare students for the more advanced mathematical topics.