Revision: geometry, counting, probability and statistics

How to use this sheet

These formulas come up across the mathematics questions, and without a calculator the fastest route is usually the formula plus a quick sanity check. Read each block, cover it, and say it back; then do the drills below without looking.

Lines in the plane

For points $(x_1, y_1)$ and $(x_2, y_2)$:

Quantity	Formula
Distance	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint	$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Slope	$m = \frac{y_2 - y_1}{x_2 - x_1}$ (for $x_1 \neq x_2$)
Line through a point with slope $m$	$y - y_1 = m(x - x_1)$
Parallel lines	$m_1 = m_2$
Perpendicular lines (neither vertical)	$m_1 m_2 = -1$
Distance from $(x_0, y_0)$ to $ax + by + c = 0$	$\frac{\lvert ax_0 + by_0 + c \rvert}{\sqrt{a^2 + b^2}}$

The line through $A(1, 2)$ and $B(3, 8)$ has slope $\frac{8 - 2}{3 - 1} = 3$, so $y - 2 = 3(x - 1)$, that is $y = 3x - 1$; both points satisfy it. The distance from $(1, 2)$ to $3x + 4y - 1 = 0$ is $\frac{\lvert 3 + 8 - 1 \rvert}{5} = 2$. A line written as $ax + by + c = 0$ has slope $-\frac{a}{b}$ when $b \neq 0$.

Circles

A circle with centre $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$. In the expanded form $x^2 + y^2 + ax + by + c = 0$:

$$\text{centre} = \left(-\frac{a}{2}, -\frac{b}{2}\right) \qquad r^2 = \frac{a^2}{4} + \frac{b^2}{4} - c$$

It is a real circle only if $r^2 > 0$, and only if $x^2$ and $y^2$ have equal coefficients and there is no $xy$ term. Completing the square gives the same results when you forget the formulas.

Parabolas

For $y = ax^2 + bx + c$ with $a \neq 0$, the vertex has $x_V = -\frac{b}{2a}$; substitute it to get $y_V$. The axis of symmetry is the vertical line $x = x_V$. The parabola opens upwards if $a > 0$ (the vertex is a minimum) and downwards if $a < 0$. For $y = 2x^2 - 8x + 5$: $x_V = \frac{8}{4} = 2$ and $y_V = 2(4) - 16 + 5 = -3$, so the vertex is $(2, -3)$, a minimum.

Areas and volumes

Figure	Formula
Triangle	$\frac{1}{2} \times \text{base} \times \text{height}$
Equilateral triangle, side $s$	$\frac{\sqrt{3}}{4}s^2$
Trapezoid	$\frac{1}{2}(b_1 + b_2)h$
Rhombus, diagonals $d_1$, $d_2$	$\frac{1}{2}d_1 d_2$
Circle	area $\pi r^2$, circumference $2\pi r$
Sector with angle $\theta$ degrees	arc $\frac{\theta}{360} \cdot 2\pi r$, area $\frac{\theta}{360} \cdot \pi r^2$
Cylinder	$V = \pi r^2 h$
Cone	$V = \frac{1}{3}\pi r^2 h$
Sphere	$V = \frac{4}{3}\pi r^3$

A sector's arc and area are the same fraction of the full circumference and the full area, because both are set by the angle.

The counting matrix

Ask one question first: does the order of the chosen items matter?

Model	Order matters?	Formula	Example
Arranging all $n$ items	yes	$n!$	5 books on a shelf: $5! = 120$
Ordered choice of $k$ from $n$	yes (ranked slots)	$\frac{n!}{(n - k)!}$	a representative and a deputy from 6: $6 \times 5 = 30$
Arranging items with repeats	yes, but identical items are interchangeable	$\frac{n!}{r_1! \, r_2! \cdots}$	the letters of BOCCONI: $\frac{7!}{2! \, 2!} = 1260$
Unordered choice of $k$ from $n$	no (a group)	$\binom{n}{k} = \frac{n!}{k!(n - k)!}$	4 delegates from 12: $\binom{12}{4} = 495$

Independent stages multiply (the product rule); separate cases that cannot happen together add (the sum rule).

Probability

With equally likely outcomes, $P(A) = \frac{\text{favourable}}{\text{total}}$. Two rules do most of the work:

• Inclusion–exclusion: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. The overlap is counted inside both $P(A)$ and $P(B)$, so subtract it once.
• Complement: $P(\text{at least one}) = 1 - P(\text{none})$. For four tosses of a fair coin, $P(\text{no heads}) = \left(\frac{1}{2}\right)^4 = \frac{1}{16}$, so $P(\text{at least one head}) = \frac{15}{16}$.

For independent events, $P(A \cap B) = P(A)\,P(B)$.

Statistics

• Work through totals: total $=$ mean $\times$ count. Class A (20 students, mean 80) and class B (30 students, mean 70) together have mean $\frac{1600 + 2100}{50} = \frac{3700}{50} = 74$, not the plain average 75 of the two means.
• Median: the middle value of the sorted data (the mean of the two middle values when the count is even). It is resistant to outliers: in a set of three or more values, making the largest value much larger moves the mean but leaves the median where it is.
• Shifting and scaling: adding $k$ to every value adds $k$ to the mean and median and leaves the range, variance and standard deviation unchanged. Multiplying every value by $k$ multiplies the mean by $k$, the standard deviation by $\lvert k \rvert$ and the variance by $k^2$.

Compound percentages

A rise of $p\%$ is the multiplier $1 + \frac{p}{100}$, a fall is $1 - \frac{p}{100}$, and successive changes multiply. Up 20% then down 20% is $1.2 \times 0.8 = 0.96$: a net fall of 4%, not 0%. To undo a change, divide by its multiplier: after a 15% discount a laptop costs €680, so the original price was $\frac{680}{0.85} = \frac{68\,000}{85} = 800$ euros (dividing top and bottom by 17 gives $\frac{4000}{5}$). Adding 15% to 680 gives 782, which is wrong.