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I'm Up and Down, and Round and Round - Definitions

Grade 9CBSE

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

🔑Concepts

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A Circle is defined as the collection of all points in a plane which are at a fixed distance from a fixed point in the plane. The fixed point is the center (OO) and the fixed distance is the radius (rr).

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A Chord is a line segment joining any two points on the circle. The Diameter (dd) is the longest chord of the circle and passes through the center.

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An Arc is a piece of a circle between two points. The longer piece is the Major Arc and the shorter piece is the Minor Arc. When the two arcs are equal, they are called Semi-circles.

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A Segment is the region between a chord and either of its arcs. These are classified into the Major Segment and the Minor Segment.

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A Sector is the region between an arc and the two radii, joining the center to the endpoints of the arc. These are classified into the Major Sector and the Minor Sector.

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A Cyclic Quadrilateral is a quadrilateral where all four vertices lie on a circle. A key property is that the sum of either pair of opposite angles is 180∘180^\circ.

📐Formulae

d=2rd = 2r

Circumference=2πr\text{Circumference} = 2\pi r

Area of Circle=πr2\text{Area of Circle} = \pi r^2

∠A+∠C=180∘ (For cyclic quadrilateral ABCD)\angle A + \angle C = 180^\circ \text{ (For cyclic quadrilateral } ABCD\text{)}

💡Examples

Problem 1:

If the radius of a circular park is 14 m14\text{ m}, calculate its diameter and its circumference. (Take π=227\pi = \frac{22}{7})

Solution:

Given r=14 mr = 14\text{ m}. Diameter d=2r=2×14=28 md = 2r = 2 \times 14 = 28\text{ m}. Circumference C=2πr=2×227×14=88 mC = 2\pi r = 2 \times \frac{22}{7} \times 14 = 88\text{ m}.

Explanation:

We use the basic definitions where the diameter is twice the radius and the circumference formula 2πr2\pi r represents the boundary of the circle.

Problem 2:

In a cyclic quadrilateral PQRSPQRS, if ∠P=110∘\angle P = 110^\circ, find the measure of the opposite angle ∠R\angle R.

Solution:

180∘−110∘70∘\begin{array}{r} 180^\circ \\ -110^\circ \\ \hline 70^\circ \end{array} ∠P+∠R=180∘\angle P + \angle R = 180^\circ 110∘+∠R=180∘110^\circ + \angle R = 180^\circ ∠R=70∘\angle R = 70^\circ

Explanation:

According to the property of cyclic quadrilaterals, the sum of opposite angles is always supplementary (180∘180^\circ).

Problem 3:

Identify the relationship between a chord of length 10 cm10\text{ cm} and a diameter of 10 cm10\text{ cm} in the same circle.

Solution:

If the diameter is 10 cm10\text{ cm} and the chord is also 10 cm10\text{ cm}, then the chord must be the diameter itself.

Explanation:

By definition, the diameter is the longest chord in a circle. If a chord's length equals the diameter's length, it must pass through the center.