Geometry Lecture Notes

Length Calculation using Pythagoras' Theorem

• Triangle 1: Calculate length AB

  • Formula: ( a^2 + b^2 = c^2 )
  • Sides: 4 and 7 (shorter sides)
  • Hypotenuse (h): ( h = \sqrt{4^2 + 7^2} = \sqrt{65} \approx 8.06 \text{ cm} )

• Triangle 2: Given hypotenuse 13, find AB

  • Sider: 12
  • Formula: ( c^2 - a^2 = b^2 )
  • Calculation: ( 13^2 - 12^2 = 25 )
  • Result: AB = ( \sqrt{25} = 5 \text{ cm} )

Angles and Parallel Lines

• Angle Identification: Using parallel lines to find angles
  • Angle Rules: Alternate angles, corresponding angles, co-interior angles

Example 1: Finding Angle X

• Identified Angles:
  • 110° (opposite is also 110°)
  • Next to 110° on straight line: 70° (angles on a straight line add to 180°)
  • Alternate angles: X = 70°
• Justification:
  • "Angles on a straight line sum to 180°"
  • "Alternate angles are equal"

Example 2: Isosceles Triangle

• Given 105° in triangle with two equal angles
  • Above 105°: 75° (angle = 180° - 105°)
  • Base angles: 75°; thus angle X = 30° (sum = 180°)
• Justification:
  • "Base angles in isosceles are equal"
  • "Alternate angles are equal"

Interior Angles of Polygons

• Hexagon:
  • Total interior angles: ( (6-2)\times180 = 720° )
  • Each angle: ( 720 / 6 = 120° )
• Octagon:
  • Total interior angles: ( (8-2)\times180 = 1080° )
  • Each angle: ( 1080 / 8 = 135° )

Angle Between Regular Octagon and Hexagon

• Using known angles:
  • Sum of angles around a point = 360°
• Calculation:
  • ( 360° - (120° + 135°) = 105° )

Circle Formulas

• Circumference: ( C = \pi \times d )
• Area: ( A = \pi r^2 )

Area of Circle Example

• Diameter: 8
• Radius: 4
• Area:
  • ( A = \pi \times 4^2 = 16\pi \approx 50.27 \text{ cm}^2 ) (rounded as per question)
• Circumference:
  • ( C = \pi \times 8 = 8\pi \approx 25.13 \text{ cm} )

Sector Area and Arc Length

• Sector Area:
  • Formula: ( A = \frac{\theta}{360} \times \pi r^2 )
  • Given: ( \theta = 102° ), Radius = 8, Area calculated as above
• Arc Length:
  • Formula: ( L = \frac{\theta}{360} \times C )

Trapezium Area

• Formula: ( A = \frac{(a + b)}{2} \times h )
• Given: a = 5, b = 9, height = 4
• Calculation: ( \frac{(5+9)}{2} \times 4 = 28 \text{ cm}^2 )

Surface Area of Cuboid

• Total areas of different visible faces:
  • Area1: 8 × 12 = 96
  • Area2 (top/bottom): 14 × 12 = 168
  • Area3: 8 × 14 = 112
• Total Surface Area: ( 2(96 + 168 + 112) = 752 \text{ cm}^2 )

Volume of Cuboid

• Volume Formula: ( V = base area \times height )
• Example:
  • Area = 8 × 12 = 96
  • Height = 14
• Volume: ( V = 96 × 14 = 1344 \text{ cm}^3 )

Volume of Cylinder

• Cylinder Volume: ( V = \pi r^2 h )
• Given Radius = 4, Height = 15
• Volume Calculation: ( V = 240\pi \approx 753.0 \text{ cm}^3 )

Surface Area of Cylinder

• Total surface area:
  • Two circles (top/bottom) + rectangle around the side
• Total: ( 152\pi \approx 477.50 \text{ cm}^2 )

Scale Factor and Similar Shapes

• Identifying scale factor between shapes, for both length and volume.

Bearing Calculation

• Bearing Rules: Measured clockwise, three digits
• Example: Find bearing from A to B = 060°.

Transformation: Vectors and Shapes

• Reflecting shapes and translating them using vectors.
  • X-coordinate and Y-coordinate impact movement direction.

Circle Theorems

• Tangents, angles at centers, and quadrilateral angles.
  • Key findings include: Opposite angles of cyclic quadrilaterals sum to 180°.
• Just overview of tangent theorems and angles formed.

Congruent Triangles

• Proving triangles are congruent through angle and side analysis; includes isosceles criteria.
• SSS and SAS criteria.

Trigonometry: Sine and Cosine Rules

• Using sine rule when opposite sides/angles known, cosine for side/angle calculations.
• Area of triangle calculation using sine: ( \frac{1}{2}ab \sin C )

Vectors

• Analyzing points on diagrams, proving points aligned via shared directional vectors.
• Matching vectors mathematically.

Summary

• Key focus on applying geometry principles to solve problems.
• Ensure understanding of relevant theorems, formulas, and relationships.