Lecture Notes: Mathematics and Geometry Concepts

Pythagoras Theorem

• Used for right-angled triangles.
• Formula: (a^2 + b^2 = c^2) where (c) is the hypotenuse.
• Example 1: Find hypotenuse using sides 4 and 7.
  • (4^2 + 7^2 = 65), (\sqrt{65} \approx 8.06).
• Example 2: Find a shorter side.
  • Given hypotenuse 13, side 12: (13^2 - 12^2 = 25), (\sqrt{25} = 5).

Angle Properties in Geometry

• Parallel lines: Alternate, Corresponding, Co-interior angles.
• Example: Find angles using given 110 degrees.
  • Use alternate angles, vertically opposite angles, and angles on a straight line.
• Isosceles triangles: Base angles are equal.
• Example: Solve angles in triangles using properties.

Polygons

• Sum of interior angles:
  • Formula: ((n-2) \times 180)
  • Hexagon (6 sides): (720) degrees, each angle (120).
  • Octagon (8 sides): (1080) degrees, each angle (135).

Circles

• Circumference: (\pi \times \text{diameter})
• Area: (\pi r^2)
• Example with radius 4 and diameter 8.

Sectors and Arc Lengths

• Area of sector: (\frac{102}{360} \times \pi r^2)
• Example using radius 8 for a sector of 102 degrees.

Miscellaneous Shapes

• Trapezium: (\frac{a+b}{2} \times \text{height})
• Surface area of cuboids: Add areas of each face.
• Volume of cuboids and cylinders: (\text{Area of base} \times , \text{height})

Similar Shapes and Scale Factors

• Scale Factor: Ratio of corresponding side lengths.
• Example: Calculate using known sides.
• Bearings: Measured clockwise from north.

Transformations

• Translation: Use vector (e.g., ([4, -2])).
• Reflection: Across lines like (x=1), (y=3), (y=x), (y=-x).
• Rotation: 90 degrees about a point.
• Enlargement: Use a scale factor from a center point.

Advanced Geometry

• Frustum: Volume of frustum equals volume of large cone minus small cone.
• Hemisphere: (\frac{4}{3} \pi r^3) for volume.
• Surface area considerations for spheres and hemispheres.

Vectors

• Midpoints and straight lines: Use vector addition and scalar multiplication.
• Example: Prove points are collinear using vectors.

Trigonometry

• Sine Rule: For non-right triangles, (\frac{a}{\sin A} = \frac{b}{\sin B}).
• Cosine Rule: (a^2 = b^2 + c^2 - 2bc \cos A) for finding unknowns in triangles.
• Area of Triangle: (\frac{1}{2} ab \sin C).

Congruent Triangles

• Show triangles are congruent using SSS (Side-Side-Side) and other congruency rules.

Circle Theorems

• Tangents, cyclic quadrilaterals, center vs. circumference angles.
• Prove and solve using circle theorems.

These notes summarize essential geometry and trigonometry concepts suitable for high school mathematics and provide examples and formulas for practical application.