Grade 9 Circle

Definition: A circle is the set of all points in a plane that are at a fixed distance from a fixed point in the plane.

Center: The fixed point is called the center of the circle.

Radius: The fixed distance from the center to any point on the circle is called the radius (plural: radii).

Chord: A line segment joining any two points on the circle is called a chord.

Diameter: A chord that passes through the center of the circle is called a diameter. The diameter is the longest chord and is equal to twice the radius (Diameter = 2 * Radius).

Arc: A continuous part of a circle is called an arc.

Tangent: A line in the plane of a circle that intersects the circle at exactly one point is called a tangent to the circle. The point where the tangent touches the circle is called the point of contact.

Secant: A line that intersects a circle at two distinct points is called a secant. A secant contains a chord of the circle.

Relationship: A tangent is a special case of a secant where the two intersection points coincide (become one).

Circumference: The distance around the circle.
Formula: Circumference (C) = 2 * pi * radius (r) or C = pi * diameter (d).

Area: The amount of surface enclosed by the circle.
Formula: Area (A) = pi * radius squared (r^2).

Value of pi: Pi (pi) is a mathematical constant approximately equal to 3.14159 or 22/7.

Example Problem: Find the circumference and area of a circle with radius 7 cm (Use pi = 22/7).

Step 1: Identify the given value. Radius (r) = 7 cm.

Step 2: Calculate Circumference using C = 2 * pi * r.
C = 2 * (22/7) * 7 = 2 * 22 = 44 cm.

Step 3: Calculate Area using A = pi * r^2.
A = (22/7) * 7 * 7 = 22 * 7 = 154 square cm.

Result: The circumference is 44 cm and the area is 154 square cm.

Theorem: The perpendicular from the center of a circle to a chord bisects the chord.

Meaning: If you draw a line segment from the center of the circle that is perpendicular (at a 90-degree angle) to a chord, that line segment will cut the chord into two equal parts.
Example: If OM is perpendicular to chord AB, where O is the center and M is a point on AB, then AM = MB.

Converse: The line joining the center of a circle to the midpoint of a chord is perpendicular to the chord. (If M is the midpoint of chord AB and O is the center, then OM is perpendicular to AB).

Theorem: The angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circle.

Meaning: If an arc forms an angle at the very center of the circle, and also forms an angle at a point on the edge of the circle (the circumference), the angle at the center will be twice as large as the angle at the circumference, as long as both angles are formed by the same arc.
Example: If arc AB subtends Angle AOB at the center O, and Angle ACB at a point C on the circumference, then Angle AOB = 2 * Angle ACB.

Special Case: The angle in a semicircle is a right angle (90 degrees). This is because a diameter subtends a straight angle (180 degrees) at the center, and half of 180 is 90.