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Circles - Define circle terminology and interpret geometric meaning of core terms

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A circle is the collection of all points in a plane which are at a fixed distance (radius) from a fixed point (centre). The distance around the boundary is the circumference.

Diagram showing a circle with its centre and radius.
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A chord is a line segment joining any two points on the circle. The diameter is the longest chord, passing through the centre, and is equal to 2×2 \times radius.

Diagram distinguishing between a chord and a diameter.
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A piece of a circle between two points is called an arc. The region between a chord and either of its arcs is called a segment (Major and Minor).

Diagram showing the division of a circle into segments by a chord.
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The region between an arc and the two radii joining the centre to the endpoints of the arc is called a sector.

Diagram showing a sector of a circle formed by two radii and an arc.

📐Formulae

d=2rd = 2r

C=2πrC = 2\pi r

C=πdC = \pi d

Area=πr2Area = \pi r^2

LengthofSemicircleArc=πrLength of Semicircle Arc = \pi r

💡Examples

Problem 1:

If the radius of a circle is 10.510.5 cm, find the length of the longest chord of the circle.

Solution:

  1. We know that the longest chord of a circle is its diameter (dd).
  2. The relationship between diameter and radius is given by d=2rd = 2r.
  3. Given r=10.5r = 10.5 cm.
  4. Substitute the value: d=2×10.5=21d = 2 \times 10.5 = 21 cm.

Explanation:

The problem asks for the longest chord, which is the definition of the diameter. By multiplying the given radius by 2, we find the length.

Problem 2:

A point PP is at a distance of 77 cm from the center of a circle with a diameter of 1010 cm. Determine if point PP lies in the interior, exterior, or on the circle.

Solution:

  1. First, find the radius (rr) of the circle: r=d2=102=5r = \frac{d}{2} = \frac{10}{2} = 5 cm.
  2. The distance of point PP from the center is given as OP=7OP = 7 cm.
  3. Compare the distance OPOP with the radius rr: Since 7>57 > 5, we have OP>rOP > r.
  4. Therefore, the point PP lies in the exterior of the circle.

Explanation:

To determine the position of a point, we compare its distance from the center to the radius. If distance >r> r, it is in the exterior; if distance <r< r, it is in the interior; if distance =r= r, it is on the circle.

Problem 3:

In a circle with centre OO and radius 55 cm, a chord ABAB is drawn. If the distance of the chord from the centre is 44 cm, find the length of the chord ABAB.

Right triangle OMA inside a circle used to find chord length.

Solution:

  1. Let OMOM be the perpendicular from the centre OO to chord ABAB. Thus OM=4OM = 4 cm.
  2. OAOA is the radius, so OA=5OA = 5 cm.
  3. In right △OMA\triangle OMA, by Pythagoras Theorem: AM2+OM2=OA2AM^2 + OM^2 = OA^2 AM2+42=52AM^2 + 4^2 = 5^2 AM2+16=25AM^2 + 16 = 25 AM2=9  ⟹  AM=3AM^2 = 9 \implies AM = 3 cm.
  4. Since the perpendicular from the centre bisects the chord, AB=2×AM=2×3=6AB = 2 \times AM = 2 \times 3 = 6 cm.

Explanation:

The perpendicular from the centre of a circle to a chord bisects the chord. This allows us to use the Pythagorean theorem on the triangle formed by the radius, the distance from the centre, and half the chord.

Problem 4:

Given a circle with centre OO, identify the major arc and minor arc if points XX and YY are on the circumference such that ∠XOY=120∘\angle XOY = 120^\circ.

Circle showing points X and Y and the central angle of 120 degrees.

Solution:

  1. The minor arc XYXY is the part of the circumference corresponding to the central angle of 120∘120^\circ.
  2. The major arc XYXY corresponds to the reflex angle ∠XOY\angle XOY.
  3. Reflex ∠XOY=360∘−120∘=240∘\angle XOY = 360^\circ - 120^\circ = 240^\circ.

Explanation:

An arc is 'minor' if the angle subtended at the centre is less than 180∘180^\circ, and 'major' if it is greater than 180∘180^\circ.