Geometry and Algebra Lecture Notes

Key Problems Discussed

1. Area of a Rectangle with Circles

   • Three circles inside a square.
   • Radius of each circle is 24 mm.
   • Width of rectangle: 4 x 24 = 96 mm.
   • Height calculation involves overlapping circles and is computed using an equilateral triangle.
   • Area of rectangle: 8600 mm² (rounded to 3 significant figures).

2. Density and Bounds Problem

   • Cuboid dimensions and mass provided.
   • Density formula: Density = Mass/Volume.
   • Calculating upper and lower bounds for dimensions and mass.
   • Final density calculated to two decimal places: 0.43.

3. Similar Shapes and Scale Factor

   • Similar triangles concept verified by tangent angles.
   • Algebraic manipulation for finding scale factor.
   • Resulting scale factor expressed in terms of variables.

4. Probability without a Tree Diagram

   • Probability calculations involving adding counters back into a bag.
   • Use of equivalent ratios to solve for the number of counters.
   • Algebraic setup to find specific counts of colored counters.

5. Bearings Calculation

   • Solving for bearings using interior angles and supplementary angle properties.
   • Use of trigonometry and cosine rule for finding specific angles.
   • Final bearing calculated as 98.6 degrees with proper notation.

6. 3D Trigonometry

   • Angle between a line and a base plane calculated using triangle properties.
   • Use of SohCahToa and cosine rule.
   • Calculation of 31.0 degrees for a specific angle.

7. Circle and Tangent Problem

   • Circle equation and a given point on the circle.
   • Gradient of radius and tangent calculated.
   • Equation of tangent derived using slope-intercept form.

8. Simultaneous Quadratic Equations

   • Solving quadratic simultaneous equations using substitution.
   • Factorization to find final solutions.
   • Different approaches to get solutions.

9. Vector Problem with Ratios

   • Vectors in geometrical shapes and calculating specific ratios.
   • Use of algebra to manipulate and solve vector equations.
   • End result focuses on specific ratio of vectors involved.

10. Complex Trigonometry and Geometry Problem

    • Problem involving congruent parallelograms and specific angles.
    • Use of cosine rule and algebraic manipulation.
    • Expression derived for specific angle's cosine value.