Scale Drawings, Bearings and Trigonometry 1. Scale Drawings Scale drawings represent real-life objects or areas at a reduced size while maintaining proportions. Scale Factors A scale is written as $\text{Drawing Distance} : \text{Real Distance}$ (e.g., $1 : 50,000$). Finding Real Distance: $\text{Real Distance} = \text{Drawing Distance} \times \text{Scale Factor}$. Finding Drawing Distance: $\text{Drawing Distance} = \text{Real Distance} \div \text{Scale Factor}$. 2. Bearings Bearings are used for navigation to specify direction. ...

Further Solving of Equations and Inequalities 1. Simultaneous Equations Two linear equations with two unknowns (usually $x$ and $y$) that must be solved simultaneously. Methods of Solution Elimination: Multiply equations so that coefficients of one variable are the same (or opposites), then add or subtract the equations to eliminate that variable. Example: Solve $2x + 3y = 13$ and $x - 2y = -4$ Multiply the second equation by 2 to match $x$ coefficients $\rightarrow 2x - 4y = -8$ Subtract the new equation from the first $\rightarrow (2x + 3y) - (2x - 4y) = 13 - (-8)$ Simplify and solve for $y \rightarrow 7y = 21 \rightarrow y = 3$ Substitute $y = 3$ back into $x - 2y = -4 \rightarrow x - 2(3) = -4 \rightarrow x = 2$ Result: $(2, 3)$ Substitution: Rearrange one equation to express one variable in terms of the other (e.g., $x = \dots$), then substitute this expression into the second equation. Example: Solve $y = 2x + 1$ and $3x + 2y = 16$ Steps: ...

Understanding Measurement 1. Standard Metric Units and Conversions Metric units follow a base-10 system. Conversion involves multiplying or dividing by powers of 10. Common Units Length: millimeter (mm), centimeter (cm), meter (m), kilometer (km) Mass: milligram (mg), gram (g), kilogram (kg), tonne (t) Capacity/Volume: milliliter (ml), liter (l), cubic centimeter ($\text{cm}^3$), cubic meter ($\text{m}^3$) Conversion Factors $1\text{ cm} = 10\text{ mm}$ $1\text{ m} = 100\text{ cm} = 1000\text{ mm}$ $1\text{ km} = 1000\text{ m}$ $1\text{ kg} = 1000\text{ g}$ $1\text{ l} = 1000\text{ ml} = 1000\text{ cm}^3$ Tip: To convert from a larger unit to a smaller unit, multiply. To convert from a smaller unit to a larger unit, divide. ...

Averages and Measures of Spread 1. Types of Averages (Central Tendency) Averages provide a single value that represents a data set. Mean The sum of all values divided by the total number of values. Non-grouped data: $\text{Mean} = \frac{\sum x}{n}$. Frequency data: $\text{Mean} = \frac{\sum (f \cdot x)}{\sum f}$ (where $f$ is frequency and $x$ is the value/midpoint). Median The middle value when data is ordered from least to greatest. ...

Pythagoras’ Theorem and Similar Shapes 1. Pythagoras’ Theorem Applies only to right-angled triangles. The Formula In a right-angled triangle with legs $a, b$ and hypotenuse $c$ (the side opposite the right angle): $$a^2 + b^2 = c^2$$ Calculations Finding the Hypotenuse: $c = \sqrt{a^2 + b^2}$. Finding a Shorter Side: $a = \sqrt{c^2 - b^2}$. 2. Similarity Two shapes are similar if they have the same shape but different sizes. ...