Lecture Notes

Right-Angled Triangles and Pythagoras' Theorem

• Length AB in Right-Angled Triangles: Use Pythagoras' Theorem (a² + b² = c²) to find missing lengths.
• Example 1: To find the hypotenuse:
  • Given sides 4 and 7: 4² + 7² = 65
  • Square root of 65 = 8.062 (hypotenuse length)
• Example 2: Given hypotenuse 13, find shorter side using subtraction:
  • 13² - 12² = 25; √25 = 5
• Key Points: Add squares for hypotenuse, subtract for shorter sides, square root to find the length.

Angles and Parallel Lines

• Parallel Line Rules: Alternate, corresponding, and co-interior angles.
• Example: Finding angles using 110° and 70° angles:
  • Alternate angles are equal; angles on a straight line add up to 180°.
• Isosceles Triangle: Base angles are equal.
• Example with Hexagon and Octagon:
  • Interior angles calculated using formula (n-2)×180.
  • Hexagon: 120° angles; Octagon: 135° angles.
  • Missing angle around a point: 360° - (sum of known angles).

Circles: Area and Circumference

• Formulas:
  • Circumference: π × diameter
  • Area: π × r²
• Example:
  • Diameter 8, radius 4 → Area = 50.27 cm²; Circumference = 25.1 cm

Trapezium Area

• Formula: (a + b)/2 × height
• Example:
  • Parallel sides 5 and 9, height 4 → Area = 28 cm²

Surface Area and Volume of Cuboids

• Surface Area: Calculate each face, double for opposite pairs.
  • Example: Faces = (8×12), (14×12), (8×14); Total SA = 752 cm²
• Volume: Area of cross-section × depth.
  • Example: Volume = 8 × 12 × 14 = 1344 cm³

Transformations

• Translation: Move by vector (x, y)
• Reflection: Reflect in line (e.g., x = 1)
• Rotation: Rotate 90° clockwise around a point
• Enlargement: Scale factor from a point (negative scale factor flips shape)

Bearings

• Rules: Clockwise from North, 3 digits
• Examples: Calculate angle movement, use corresponding angles

Circle Theorems and Geometry

• Tangents and Radii: Tangents from a point are equal; meet radii at 90°
• Cyclic Quadrilateral: Opposite angles add up to 180°
• Congruent Triangles: Use SSS, SAS, ASA for proof

Sine and Cosine Rules

• Sine Rule: For non-right triangles with opposite pairs
• Cosine Rule: When no opposite pairs are available

Vectors

• Concept: Show common direction for collinearity using scalar multiples

Sector Area and Triangle

• Sector Area: Part of circle's area; use fraction of 360°
• Triangle within Sector: Use 1/2abSinC for area

Note: This is a high-level summary of the lecture content on mathematical geometry, focusing especially on triangles, circles, and various mathematical theorems and formulas related to these shapes.