Lecture Notes: Geometry and Algebra Concepts

Circles and Rectangle Problem

• Circles in a Square: Three circles with a radius of 24mm each, inside a rectangle.
• Width of Rectangle: Calculated as 4 radii = 96mm.
• Height Calculation: Involves overlap in circles' radii.
  • Use equilateral triangle formed by circle centers.
  • Calculate triangle's height (x) using trigonometry: 48 * sin(60°) = 41.5mm.
• Total Height: Add two radii to the height: 89.569mm.
• Area of Rectangle: Width * Height = 96 * 89.569 = 8,598.624 mm².
  • Rounded to 3 significant figures: 8,600 mm².*

Density and Bounds

• Cuboid Dimensions: Length, width, height provided.
• Mass: 1970 to the nearest five.
• Density Formula: Density = Mass/Volume.
• Error Intervals: Calculate bounds for each dimension.
  • Calculate upper and lower bounds for volume.
• Final Density: Rounded to two decimal places: 0.43.

Similar Shapes and Algebra

• Given: Triangle with tangential properties, similar shapes.
• Algebraic Fraction Setup: Solve by cross-multiplying.
• Quadratic Solving: Factor to find x = 5/3.

Probability and Algebraic Fractions

• Probability Problem: Focus on ratios and creating algebraic equations.
• Ratio Setup: Used to determine equivalent ratios between colors (red and green).
• Equation Solving: Use algebra to find the number of counters.

Bearings and Trigonometry

• Bearing Calculation: From one point to another.
• Co-interior Angles: Add up to 180°.
• Cosine Rule & Sine Rule: Used to find unknown angles and sides.
• Resultant Bearing: 098.6° with three significant figures.

3D Trigonometry

• Triangle Analysis: Use of Sokotoa for height calculations.
• Pythagoras: To determine base lengths.
• Final Angle Calculation: 31.0° using trigonometry.

Circle Equation and Tangents

• Equation of Tangent: Find gradient from the center to a point.
• Gradient Calculation: Use negative reciprocal for tangent.
• Line Equation: y = mx + c, find c through substitution.

Simultaneous Equations

• Quadratic and Linear Equations: Rearrange and substitute.
• Factorizing Quadratics: Simplify where possible before solving.
• Solutions for x and y: Found through substitution.

Vector Questions

• Ratio and Proportions: Work with vectors and midpoints.
• Vector Calculation: To find ratios like k and n.
• Straight Line Analysis: Ensure vectors align correctly.

Advanced Vector and Trigonometry Problems

• Complex Vector Calculations: Focus on geometrical relationships.
• Cosine Rule Application: In context of given shapes.
• Final Cosine Expression: Simplified to provided form.