NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.1

Definitions of various terms related to circle

Circle and Related Definitions

Definitions of various terms related to circle

Q-1 Fill in the blanks

  1. The centre of a circle lies in __________ of the circle. (exterior/ interior)

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

  3. The longest chord of a circle is a __________ of the circle.

  4. An arc is a __________ when its ends are the ends of a diameter.

  5. Segment of a circle is the region between an arc and __________ of the circle.

  6. A circle divides the plane, on which it lies, in __________ parts.

Solution:

  1. The centre of a circle lies in interior of the circle.

  2. A point, whose distance from the centre of a circle is greater than its radius lies in exterior of the circle.

  3. The longest chord of a circle is a diameter of the circle.

    Of all the chords of a circle diameters are longest

    Among Several Chords Diameter Is Longest

    Of all the chords of a circle diameters are longest

  4. An arc is a semi-circle when its ends are the ends of a diameter.

    A semicircle ends in diameter

    A Semicircle

    A semicircle ends in diameter

  5. Segment of a circle is the region between an arc and chord of the circle.

  6. A circle divides the plane, on which it lies, in three parts. The circle itself, its interior and exterior.

    Circle, its interior and exterior

    Circle, Interior and Exterior

    Circle, its interior and exterior

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

  1. Line segment joining the centre to any point on the circle is a radius of the circle.

  2. A circle has only finite number of equal chords.

  3. If a circle is divided into three equal arcs, each is a major arc.

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

  5. Sector is the region between the chord and its corresponding arc.

  6. A circle is a plane figure.

Solution (i):

  • True, all the points on the circle are equidistant from the centre of the circle, and this equal distance is the radius of the circle.

Solution (ii):

  • False. For example, in the below figure we have two chords CD and AB which are both the same length.

    Chors of a circle with same length

    Equal Chords of a Circle

    Chors of a circle with same length

    Here OM and ON are perpendicular to AB and CD respectively. ON and OM are equal. All the chords of the circle which have same perpendicular distance from the center will be equal. Clearly there are infinite such chords.

Solution (iii):

  • False. Following figure shows a major arc and a minor arc:

    Major arc > 180 degrees and minor arc is less than 180 degrees

    Major Arc and Minor Arc

    Major arc > 180 degrees and minor arc is less than 180 degrees

  • An arc of the circle which is greater than 180 is called major otherwise they it is called minor arc.

  • False, in the figure below, PQ, QR, and PR are equal parts of a circle, they each are minor arcs since they subtend an angle less than 180 degree at the center.

Circle c Circle c: Circle through Q with center A Point Q Q = (2.32, 2.18) Point Q Q = (2.32, 2.18) Point Q Q = (2.32, 2.18) Point R Point R: Point on c Point R Point R: Point on c Point R Point R: Point on c Point P Point P: Point on c Point P Point P: Point on c Point P Point P: Point on c S text1 = "S"

Circle Points Are P, Q, and R

Circle points are P, Q, R also arc RSQ, QPR is a major arc.

Solution (iv):

  • True. Let PQ be a chord which is twice as long as its radius. The longest chord of a circle can be 2 times the radius, i.e. the length of the diameter. All other chords will be smaller than diameter. Therefore the only possible chord twice the radius is diameter and it this chord will be passing through the centre of the circle.

Circle c Circle c: Circle through Q with center O Segment f Segment f: Segment [Q, P] Point O O = (0.56, 2.08) Point O O = (0.56, 2.08) Point O O = (0.56, 2.08) Point Q Q = (2.74, 3.94) Point Q Q = (2.74, 3.94) Point Q Q = (2.74, 3.94) Point P Point P: Point on c Point P Point P: Point on c Point P Point P: Point on c r text1 = "r" r text2 = "r"

Circle and Chord of PQ

Circle and chord of PQ passing through center O

Solution (v):

  • False. Sector is the region between an arc and two radii joining the centre to the end points of the arc.

  • For example, in below figure, OPQ is the sector of the circle.

Circle c Circle c: Circle through B with center A Segment f Segment f: Segment [C, A] Segment g Segment g: Segment [A, B] Point A A = (0.52, 2.6) Point A A = (0.52, 2.6) Point A A = (0.52, 2.6) Point B B = (1.9, 0.76) Point B B = (1.9, 0.76) Point B B = (1.9, 0.76) Point C Point C: Point on c Point C Point C: Point on c Point C Point C: Point on c Sector text2 = "Sector"

the Circle and Sector

The circle and sector of OPQ on circle

Solution (vi):

True. A circle is a two-dimensional figure and it can also be referred to as a plane figure. It can be drawn on a plane (a paper to be approximate)

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