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. ...
Straight Lines and Quadratic Equations 1. Straight Line Graphs Represented on a 2D Cartesian coordinate system $(x, y)$. The Equation of a Line General form: $y = mx + c$ $m$: Gradient (slope). $c$: $y$-intercept (where the line crosses the $y$-axis). Gradient ($m$) Gradient measures the steepness: $\text{gradient} = \frac{\text{vertical change}}{\text{horizontal change}} = \frac{\text{rise}}{\text{run}}$. Calculating Gradient from Two Points: Given $(x_1, y_1)$ and $(x_2, y_2)$: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ ...
Sequences, Surds, and Sets 1. Sequences A sequence is an ordered list of numbers following a specific rule. Term-to-Term Rules Describes how to get from one term to the next. Arithmetic Sequence: Constant difference added or subtracted (e.g., $+3, +3, +3$). Geometric Sequence: Constant ratio multiplied or divided (e.g., $\times 2, \times 2, \times 2$). The $n$-th Term ($u_n$) The formula to find any term in the sequence based on its position $n$. ...
Introduction to Probability 1. Basic Probability Probability Scale Probability is measured on a scale from $0$ to $1$: $0$: Impossible event. $1$: Certain event. $0.5$: Equally likely as not. Fundamental Rules Complementary Events: The probability of an event NOT happening is $P(\text{not } A) = 1 - P(A)$. Relative Frequency: An estimate of probability based on experimental data. $$\text{Relative Frequency} = \frac{\text{Number of successful trials}}{\text{Total number of trials}}$$ Expected Frequency: The number of times an event is expected to occur over a number of trials. $$\text{Expected Frequency} = P(A) \times \text{Total Trials}$$ 2. Sample Space A Sample Space is a list of all possible outcomes of an experiment. ...
Perimeter, Area, and Volume 1. 2D Shapes Area and Perimeter Shape Perimeter Area Rectangle $2(l + w)$ $l \times w$ Triangle - $\frac{1}{2} \times \text{base} \times \text{height}$ Parallelogram - $\text{base} \times \text{height}$ Trapezium - $\frac{1}{2}(a + b)h$ Circles Circumference: $C = 2\pi r$ or $\pi d$. Area: $A = \pi r^2$. Arc Length: $L = \frac{\theta}{360} \times 2\pi r$. Sector Area: $A = \frac{\theta}{360} \times \pi r^2$. For major sectors, $\theta$ is the reflex angle. ...