FORM THREE MATHEMTICS STUDY NOTES TOPIC 6:CIRCLES

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FORM THREE MATHEMTICS STUDY NOTES TOPIC 6:CIRCLES

FORM THREE MATHEMTICS STUDY NOTES TOPIC 6:CIRCLES



CIRCLES

Definition of Terms
Circle, Chord, Radius, Diameter, Circumference, Arc, Sector, Centre and Segment of a Circle
Define circle, chord, radius, diameter, circumference, arc, sector, centre and segment of a circle
circle: is the locus or the set of all points equidistant from a fixed point called the center.
Arc: a curved line that is part of the circumference of a circle
Chord: a line segment within a circle that touches 2 points on the circle.
Circumference: The distance around the circle.
Diameter: The longest distance from one end of a circle to the other.
Origin: the center of the circle
Pi(π):A number, 3.141592…, equal to (the circumference) / (the diameter) of any circle.
Radius: distance from center of circle to any point on it.
Sector: is like a slice of pie (a circle wedge).
Tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle.
NB: Diameter = 2 x radius of circle
Circumference of Circle = PI x diameter = 2 PI x radius
A Tangent to a Circle
Describe a tangent to a circle
Tangent is a line which touches a circle. The point where the line touches the circle is called the point of contact. A tangent is perpendicular to the radius at the point of contact.
Tangent Properties of a Circle
Identify tangent properties of a circle
A tangent to a circle is perpendicular to the radius at the point of tangency. A common tangent is a line that is a tangent to each of two circles. A common external tangent does not intersect the segment that joins the centers of the circles. A common internal tangent intersects the segment that joins the centers of the circles.
Tangent Theorems
Prove tangent theorems
Theorem 1
If two chords intersect in a circle, the product of the lengths of the segments of one chord equal the product of the segments of the other.
Intersecting Chords Rule: (segment piece)×(segment piece) =(segment piece)×(segment piece)
Theorem Proof:
Theorem 2:
If two secant segments are drawn to a circle from the same external point, the product of the length of one secant segment and its external part is equal to the product of the length of the other secant segment and its external part.
Secant-Secant Rule: (whole secant)×(external part) =(whole secant)×(external part)
Theorem 3:
If a secant segment and tangent segment are drawn to a circle from the same external point, the product of the length of the secant segment and its external part equals the square of the length of the tangent segment.
Secant-Tangent Rule:(whole secant)×(external part) =(tangent)2
Theorems Relating to Tangent to a Circle in Solving Problems
Apply theorems relating to tangent to a circle in solving problems
Example 7
Two common tangents to a circle form a minor arc with a central angle of 140 degrees. Find the angle formed between the tangents.
Solution
Two tangents and two radii form a figure with 360°. If y is the angle formed between the tangents then y + 2(90) + 140° = 360°
y = 40°.

The angle formed between tangents is 40 degrees.

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