Revision: Circle Mathematics (English Medium) ICSE Class 9 CISCE

A circle is a closed curve where all points on the boundary (called the circumference) are at the same distance from a fixed point inside it.

• The fixed point inside the circle is called the center (O)

The radius is a straight line segment that connects the center of the circle to any point on its circumference.

Characteristics:

• Symbol: Usually represented as r

• All radii of a circle have the same length

• A circle has infinite radii (one to every point on the circumference)

• The radius is always half the diameter

• Radius = `"Diameter"/"2"`

 The diameter is a straight line segment that passes through the center of the circle and has both endpoints on the circumference.

Characteristics:

• The diameter passes through the center

• A circle has infinite diameters

• The diameter is the longest possible chord of a circle

• The diameter is twice the radius

• Diameter = 2 × Radius and

A chord is a straight line segment that connects any two points on the circumference of the circle.

Characteristics:

• A circle has infinite chords

• The diameter is the longest chord in any circle

• Chords closer to the centre are longer than chords farther from the center

The line segment, joining any two points on the circumference of the circle, is called a chord. 

We have to prove that

AQ = `1/2` (perimeter of ΔABC)

Perimeter of ΔABC = AB + BC + CA

= AB + BP + PC + CA

= AB + BQ + CR + CA

(∵ Length of tangents from an external point to a circle are equal ∴ BP = BQ and PC = CR)

= AQ + AR  ...(∵ AB + BQ = AQ and CR + CA = AR)

= AQ + AQ  ...(∵ Length of tangents from an external point are equal)

= 2AQ

⇒ AQ = `1/2` (Perimeter of ΔABC)

Hence proved.

Let ABCD be a parallelogram inscribed in a circle.

Now, ∠BAD + ∠BCD

(Opposite angles of a parallelogram are equal.)

And ∠BAD + ∠BCD = 180°

(A pair of opposite angles in a cyclic quadrilateral are supplementary.)

∠BAD + ∠BCD = `(180^circ)/2` = 90°

The other two angles are 90°, and the opposite pair of sides are equal.

∴ ABCD is a rectangle.

Statement:
The angle in a semicircle is a right angle.

Result:

∠ACB = 90^∘

Short Proof (Idea):

• Diameter subtends an angle of 180° at the centre.

• The angle at the circle is half of it.

• Therefore, angle = 90°.

Statement:
Angles in the same segment of a circle are equal.

Result:

∠ACB = ∠ADB

Short Proof (Idea):

• Both angles stand on the same arc.

• The angle at the centre is double each of them.

• Hence, both angles are equal.

Statement:
The angle subtended by an arc at the centre of a circle is double the angle subtended by it at any point on the remaining part of the circle.

Result:

∠AOB = 2∠ACB

Short Proof (Idea):

• Join the centre to the points on the circle.

• Radii form isosceles triangles.

• Angle at centre = sum of angles at the circle.

• Hence, the angle at the centre is double the angle at the circle.

Statement:
If an arc of a circle subtends a right angle at any point on the remaining part of the circle, then the arc is a semicircle.

Result:
Arc AB is a semicircle
(or AB is a diameter)

Short Proof (Idea):

• Given angle at the circle ∠ACB = 90°.

• The angle at the centre is double the angle at the circle.

• Therefore, ∠AOB = 2 × 90° = 180°.

• Hence, A, O, and B lie on a straight line, so AB is a diameter.

• Therefore, arc AB is a semicircle.

• Diameter → Longest chord of a circle

• Perpendicular from centre to a chord → Bisects the chord

• Line joining centre to midpoint of a chord → Perpendicular to the chord

• The greater the chord → Nearer to the centre

• The smaller the chord, → farther from the centre

• Equal chords → Equidistant from the centre

• Chords equidistant from centre → Equal in length

• Only one circle passes through three non-collinear points

1. Arc Definition: An arc is a curved portion of a circle's circumference between two points.

2. Two Types: Minor arc (< 180°) and Major arc (> 180°).

3. Semicircle: When the arc angle is exactly 180°, it's called a semicircle.

4. Complete Circle: Minor arc + Major arc = 360° (complete circumference).

Definition: A segment is a region of a circle bounded by a chord and its arc

Two Main Types:

• Minor Segment = smaller piece

• Major Segment = larger piece

Semicircle Special Case:

• Formed when chord = diameter

• Creates two perfectly equal segments

• Each semicircle = half the circle's area