Circles and Cylinders Notes

Key Terms of Circles

• Centre: The middle point of a circle from which all points on the circumference are equidistant.
• Radius: The distance from the centre to any point on the circle.
• Chord: A line segment with both endpoints on the circumference of the circle.
• Diameter: A chord that passes through the centre; it is twice the length of the radius ($d = 2r$).
• Circumference: The distance around the circle, calculated as $C = 2\pi r$.
• Tangent: A straight line that touches the circle at exactly one point.
• Arc: A part of the circumference of a circle.
• Sector: The area enclosed by two radii and the arc between them.
• Segment: The area between a chord and the arc it subtends.

Properties of Chords and Tangents

• Tangent Definition: A tangent touches a circle at a single point referred to as the point of contact.
• Chord Definition: A chord connects two points on the circumference of the circle.

Circle Theorems

1. Tangent Perpendicularity: A tangent to a circle is perpendicular to the radius at the point of contact. (Illustration: Radius $OX$ perpendicular to tangent $AB$)
2. Tangents from External Point: Tangents drawn to a circle from a single external point are equal in length, i.e., if $A$ is the external point, then $AX = AY$ where $X$ and $Y$ are the points of contact.
3. Angle Bisector from External Point: The line from an external point to the centre of the circle bisects the angle formed by the two tangents, i.e., $\angle ZOAX = \angle ZOAY$.
4. Radius Bisecting Chord: A radius of the circle bisects any chord at a right angle. If $O$ is the centre and $M$ is the midpoint of chord $BC$, then $\angle BMO = 90°$ and $BM = CM$.

Practical Examples

• Example Calculation: Given a circle where radius $OA = 5 cm$ and tangent $AB = 12 cm$, calculate the length of segment $OB$ using Pythagorean theorem:
  $OB^2 = OA^2 + AB^2$
  $OB^2 = 5^2 + 12^2\Rightarrow OB = \sqrt{25 + 144} = \sqrt{169} = 13 cm$.

Exercises

• Given tangents $TP$ and $TQ$ to a circle, find the unknown angles or lengths using the properties identified above.
• Various diagrams can show different tangent scenarios for solving for $x$, $y$, etc.

Examples and Diagrams

• In diagrams showing circle properties, often angles such as $70°$ or $52°$ are used to express relationships between tangents and chords.
• For example, using properties, if given a radius of $7 cm$ and $12 cm$ for two concentric circles with the tangent cutting through, one must find the segment length of $AB$ where the tangent meets the outer circle.

Surface Area of a Cylinder

• To find the surface area, refer to supplementary materials or presentations on the proportion relating circumference and height to find lateral and base areas of cylinders.