Glossary
Every technical word in the course and what it means in plain language, with flashcards for practice.
Every technical term in the course, with a plain-language meaning. On every page these words are marked with a dotted underline: hover or tap one to see its meaning. The “Used in” links point to the sections where each term appears.
Ch 1 · Real NumbersCh 2 · PolynomialsCh 3 · Pair of Linear Equations in Two VariablesCh 4 · Quadratic EquationsCh 5 · Arithmetic ProgressionsCh 6 · TrianglesCh 7 · Coordinate GeometryCh 8 · Introduction to TrigonometryCh 9 · Some Applications of TrigonometryCh 10 · CirclesCh 11 · Areas Related to CirclesCh 12 · Surface Areas and VolumesCh 13 · StatisticsCh 14 · Probability
Ch 1 · Real Numbers
| Term | Meaning | Used in |
|---|---|---|
| co-prime | Two numbers are co-prime if their HCF is $1$, i.e. they share no prime factor, e.g. $8$ and $15$. | 1.2 1.3 |
| common factor | A number that divides each of the given numbers exactly. $1$ is a common factor of any numbers. | 1.1 1.2 1.3 4.1 |
| common multiple | A number that is a multiple of each of the given numbers; $48$ is a common multiple of $6$ and $8$, and the smallest one ($24$) is the LCM. | 1.2 |
| composite number | A natural number greater than $1$ that has a factor other than $1$ and itself, such as $4, 9, 91$. The number $1$ is neither prime nor composite. | 1.1 |
| division ladder | A way to factorise: keep dividing by the smallest prime that goes in exactly until you reach $1$; the divisors are the prime factors. | 1.1 |
| exponent | The power to which a number is raised. In $2^3$ the exponent is $3$, meaning $2\times2\times2$. | 1.1 1.2 |
| factor tree | A diagram that splits a number into two factors, then splits each composite factor again, until every branch ends in a prime. | 1.1 |
| Fundamental Theorem of Arithmetic | Every composite number can be written as a product of primes, and this factorisation is unique apart from the order of the factors. | 1.1 1.3 |
| HCF (highest common factor) | The largest natural number that divides each of the given numbers exactly, e.g. $\operatorname{HCF}(84,120)=12$. | 1.1 1.2 1.3 |
| irrational number | A real number that cannot be written as $\tfrac pq$ with integers $p$, $q$ and $q\ne0$, such as $\sqrt2$, $\sqrt3$ and $3+2\sqrt5$. | 1.3 |
| LCM (lowest common multiple) | The smallest natural number that is a multiple of each of the given numbers, e.g. $\operatorname{LCM}(84,120)=840$. | 1.1 1.2 3.3 |
| lowest terms | A fraction $\tfrac pq$ is in lowest terms when $p$ and $q$ are co-prime, e.g. $\tfrac{84}{120}=\tfrac{7}{10}$. Divide top and bottom by their HCF to get there. | 1.3 8.1 14.1 14.2 |
| perfect square | A whole number that is the square of a whole number, such as $1, 4, 9, 16, 25$. In its prime factorisation every exponent is even, and its square root is rational. | 1.1 1.3 4.3 14.2 |
| prime factorisation | Writing a number as a product of primes, usually in increasing order with powers, e.g. $360=2^3\times3^2\times5$. | 1.1 1.2 1.3 |
| prime number | A natural number greater than $1$ whose only factors are $1$ and itself, such as $2, 3, 5, 7, 11$. | 1.1 1.2 1.3 14.1 |
| proof by contradiction | A way of proving a statement: assume it is false, reason correctly until you reach something impossible, and conclude that the statement must be true. | 1.3 |
| rational number | A number that can be written as $\tfrac pq$ with integers $p$, $q$ and $q\ne0$, such as $\tfrac34$, $-5$ or $\sqrt{49}=7$. | 1.3 |
Ch 2 · Polynomials
| Term | Meaning | Used in |
|---|---|---|
| coefficient | The number multiplying a power of the variable. In $3x^2-5x+\sqrt2$ the coefficients are $3$, $-5$ and $\sqrt2$ (the last one is the constant term). | 2.1 2.2 3.1 3.2 |
| cubic polynomial | A polynomial of degree $3$, written $ax^3+bx^2+cx+d$ with $a\ne0$. It has at least one and at most three zeroes. | 2.1 2.2 |
| degree of a polynomial | The highest power of the variable that appears with a non-zero coefficient. $5x^3-2x^7+1$ has degree $7$; a non-zero constant has degree $0$. | 2.1 |
| Factor Theorem | If $p(\alpha)=0$ for a polynomial $p(x)$, then $x-\alpha$ is a factor of $p(x)$ (and conversely). | 2.2 |
| leading coefficient | The coefficient of the highest power of the variable. In $6x^2-7x-3$ it is $6$; it is the number $a$ in $ax^2+bx+c$. | 2.2 |
| linear polynomial | A polynomial of degree $1$, written $ax+b$ with $a\ne0$. It has exactly one zero, $-\tfrac ba$, and its graph is a straight line. | 2.1 |
| parabola | The U-shaped curve that is the graph of a quadratic polynomial. It opens upwards (∪) when $a\gt0$ and downwards (∩) when $a\lt0$. | 2.1 2.2 4.1 4.2 |
| polynomial | An expression such as $4x^3-2x+9$ made of terms $a x^n$ where every power $n$ of the variable is a whole number $0, 1, 2, \dots$ Expressions like $\sqrt x$ or $\tfrac1x$ are not polynomials. | 2.1 2.2 4.1 4.2 |
| product of zeroes | For a quadratic $ax^2+bx+c$ ($a\ne0$) with zeroes $\alpha$, $\beta$: $\;\alpha\beta=\tfrac ca$. | 2.2 |
| quadratic polynomial | A polynomial of degree $2$, written $ax^2+bx+c$ with $a\ne0$. Its graph is a parabola, and it has at most two zeroes. | 2.1 2.2 4.1 4.2 |
| reciprocal | The reciprocal of a non-zero number $x$ is $\tfrac1x$; a number and its reciprocal multiply to $1$ (for example $3$ and $\tfrac13$). | 2.2 4.2 4.3 7.2 |
| splitting the middle term | A way to factorise $ax^2+bx+c$: write $bx$ as the sum of two terms whose coefficients add to $b$ and multiply to $ac$, then group. For example $6x^2-7x-3=6x^2-9x+2x-3=(3x+1)(2x-3)$. | 2.2 4.2 4.3 |
| sum of zeroes | For a quadratic $ax^2+bx+c$ ($a\ne0$) with zeroes $\alpha$, $\beta$: $\;\alpha+\beta=-\tfrac ba$. | 2.2 |
| zero of a polynomial | A number $k$ for which $p(k)=0$. On the graph of $y=p(x)$ the zeroes are the $x$-coordinates of the points where the graph meets the $x$-axis. | 2.1 4.1 |
Ch 3 · Pair of Linear Equations in Two Variables
| Term | Meaning | Used in |
|---|---|---|
| back-substitution | The last step of the substitution method: putting the value just found (e.g. $x=4$) back into the expression for the other variable (e.g. $y=7-x=3$). | 3.2 |
| coincident lines | Two lines that lie exactly on top of each other: every point of one is a point of the other. | 1.2 3.1 3.2 13.3 |
| consistent pair | A pair of equations with at least one solution: the lines intersect (one solution) or coincide (infinitely many). | 3.1 |
| dependent pair | A consistent pair whose two equations describe the same line, so it has infinitely many solutions; e.g. $x-2y=-2$ and $3x-6y=-6$. | 3.1 |
| downstream | Moving in the same direction as the current of a river. A boat's speed downstream is (its speed in still water) $+$ (speed of the stream). | 3.3 4.2 |
| eliminate | To make a variable disappear from an equation, e.g. adding $x+y=7$ and $x-y=1$ eliminates $y$ and leaves $2x=8$. | 3.3 5.2 9.2 |
| elimination method | Solving a pair of linear equations by multiplying them so that one variable has equal (or opposite) coefficients, then subtracting (or adding) the equations so that this variable cancels. | 3.3 |
| general form (of a linear equation) | Writing a linear equation with everything on one side, as $ax+by+c=0$; e.g. $3x=7-2y$ becomes $3x+2y-7=0$. | 3.1 5.1 |
| graphical method | Solving a pair of linear equations by drawing both lines on one graph and reading the point where they meet. | 3.1 |
| inconsistent pair | A pair of equations with no solution: the two lines are parallel, e.g. $x+y=2$ and $x+y=5$. | 3.1 3.2 |
| linear equation in two variables | An equation that can be written as $ax+by+c=0$ with $a$, $b$ not both zero, e.g. $2x+3y=12$. Its graph is a straight line. | 3.1 |
| pair of linear equations | Two linear equations in the same two variables, studied together: $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$. | 3.1 3.2 3.3 |
| simultaneous equations | Another name for a pair of equations that must be true at the same time, such as $x+y=7$ and $x-y=1$. | 3.2 |
| speed in still water | The speed a boat (or swimmer) would have on water that is not flowing; the current adds to it downstream and subtracts from it upstream. | 3.3 4.2 |
| subject of an equation | The variable that stands alone on one side, e.g. $y$ in $y=7-x$. Making $y$ the subject of $x+y=7$ means rewriting it as $y=7-x$. | 3.2 |
| substitution method | Solving a pair of linear equations by writing one variable in terms of the other from one equation, putting that expression into the other equation, and solving the resulting equation in one variable. | 3.2 |
| two-digit number | A number from $10$ to $99$. With tens digit $x$ and units digit $y$ it equals $10x+y$; with the digits reversed it is $10y+x$. | 3.3 4.2 5.3 |
| unique solution | Exactly one solution. A pair of linear equations has a unique solution when its two lines cross at one point. | 3.1 3.2 3.3 |
| upstream | Moving against the current of a river. A boat's speed upstream is (its speed in still water) $-$ (speed of the stream). | 3.3 4.2 |
Ch 4 · Quadratic Equations
| Term | Meaning | Used in |
|---|---|---|
| discriminant | The number $D=b^2-4ac$ for the equation $ax^2+bx+c=0$. Its sign tells you how many real roots there are: two if $D\gt0$, one repeated root if $D=0$, none if $D\lt0$. | 4.3 |
| distinct real roots | Two different real numbers that both satisfy the equation. A quadratic equation has them exactly when $D=b^2-4ac\gt0$; its parabola cuts the $x$-axis twice. | 4.3 |
| factor pair | Two whole numbers whose product is a given number. The factor pairs of $12$ are $1\times12$, $2\times6$ and $3\times4$; we search them when splitting the middle term. | 4.2 |
| factorisation method | Solving $ax^2+bx+c=0$ by writing the left side as a product of two linear factors (usually by splitting the middle term) and setting each factor equal to $0$. | |
| linear factor | A factor of the form $px+q$ with $p\ne0$, such as $2x+1$. Each linear factor of $ax^2+bx+c$ gives one root, $x=-\tfrac qp$. | |
| mathematical model | An equation (or set of equations) that describes the conditions of a real situation, such as $x(2x+3)=90$ for a garden of area $90$ m² whose length is $2x+3$. | 3.2 4.1 |
| nature of roots | Whether a quadratic equation has two distinct real roots, two equal real roots (a repeated root) or no real roots — decided by the sign of the discriminant. | 4.3 |
| no real roots | No real number satisfies the equation. For $ax^2+bx+c=0$ this happens exactly when $D=b^2-4ac\lt0$; the parabola never meets the $x$-axis. | 4.1 4.3 |
| quadratic equation | An equation that can be written as $ax^2+bx+c=0$ with $a\ne0$, for example $2x^2+3x-90=0$. The unknown appears squared, and there are at most two real roots. | 2.2 4.1 4.2 4.3 |
| quadratic formula | The formula $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ that gives the roots of any quadratic equation $ax^2+bx+c=0$ ($a\ne0$) whenever $b^2-4ac\ge0$. | 4.1 4.3 5.3 |
| repeated root | A root that appears twice because both factors are the same, as in $(3x-2)^2=0$, whose roots are $\tfrac23,\tfrac23$. The parabola just touches the $x$-axis there. | 4.2 4.3 |
| root of an equation | A number that makes the equation true when it is put in place of the unknown. $3$ is a root of $x^2-5x+6=0$ because $9-15+6=0$. | 4.1 4.3 |
| satisfy (an equation) | A value satisfies an equation if substituting it makes both sides equal. The roots of an equation are the values that satisfy it. | 3.1 3.2 3.3 4.1 |
| standard form (of a quadratic equation) | The tidy way of writing a quadratic equation: $ax^2+bx+c=0$, with every term on the left in descending powers of $x$ and $0$ on the right. | 4.1 4.2 4.3 |
| surd form | Writing an irrational answer exactly with square roots in their simplest form, such as $-2\pm\sqrt5$ or $\dfrac{2\pm\sqrt7}{3}$, instead of as a rounded decimal. | 4.3 |
| zero-product rule | If a product of two numbers is $0$, at least one of the numbers is $0$. So $(x-5)(x+2)=0$ means $x-5=0$ or $x+2=0$. | 4.2 |
Ch 5 · Arithmetic Progressions
| Term | Meaning | Used in |
|---|---|---|
| arithmetic mean | The ordinary average. For two numbers $a$ and $c$ it is $\tfrac{a+c}{2}$; three numbers $a, b, c$ are in AP exactly when $b$ is the arithmetic mean of $a$ and $c$, i.e. $2b=a+c$. | 5.1 |
| arithmetic progression | A list of numbers in which every term after the first is obtained by adding the same fixed number $d$ to the term before it, e.g. $3, 7, 11, 15, \dots$ Shortened to AP. | 5.1 5.2 5.3 6.3 |
| arithmetic series | The terms of an AP joined by plus signs, such as $3+7+11+15$. Its value is the sum $S_n$ of those terms. | 5.3 |
| common difference | The fixed number $d$ added to each term of an AP to get the next one: $d=a_2-a_1=a_3-a_2=\cdots$ (later term minus earlier term). It can be positive, negative or zero. | 5.1 5.2 5.3 |
| finite AP | An AP with a limited number of terms; it has a last term $l$, e.g. the dates of the Mondays in a month. | 5.1 5.2 |
| first negative term | In a decreasing AP, the earliest term below $0$. Solve $a+(n-1)d\lt0$ and take the smallest whole number $n$ that works. | 5.2 |
| general form of an AP | The way every AP can be written from its first term $a$ and common difference $d$: $a,\ a+d,\ a+2d,\ a+3d,\ \dots$ | 5.1 |
| general term | The $n$th term of an AP written as a formula in $n$: $a_n=a+(n-1)d$. Putting $n=1,2,3,\dots$ gives the terms one by one. | 5.2 |
| infinite AP | An AP that goes on for ever, with no last term, e.g. the odd numbers $1, 3, 5, 7, \dots$ | 5.1 |
| instalment | One of several part-payments by which a loan or a price is paid back. When each instalment rises or falls by a fixed amount, the instalments form an AP and their total is a sum $S_n$. | 5.3 |
| last term | The final term $l$ of a finite AP. If the AP has $m$ terms, $l=a+(m-1)d$. | 5.1 5.2 5.3 |
| middle term of an AP | In a finite AP with an odd number $m$ of terms, the term in the centre: the $\tfrac{m+1}{2}$th term. It equals the arithmetic mean of the first and last terms. | 5.2 |
| number of terms | How many terms a finite AP has. Solve $l=a+(m-1)d$ for $m$; e.g. $7, 13, \dots, 205$ has $m=34$ terms. | 5.1 5.3 |
| pairing trick | Adding an AP by pairing the first term with the last, the second with the second-last, and so on. Every pair adds up to $a+l$, so $n$ terms add up to $\tfrac n2(a+l)$. Often credited to Gauss, who used it to find $1+2+\dots+100=5050$. | 5.3 |
| sum of an AP | The total $S_n$ of the first $n$ terms of an AP, $S_n=a_1+a_2+\dots+a_n$. It is found in one step with $S_n=\tfrac n2\left[2a+(n-1)d\right]=\tfrac n2(a+l)$. | 5.3 |
| term from the end | A term of a finite AP counted backwards from the last term $l$. The $n$th term from the end is $l-(n-1)d$, the same as the $(m-n+1)$th term from the start when there are $m$ terms. | 5.2 5.3 |
| triangular number | A sum $1+2+3+\dots+n=\tfrac{n(n+1)}{2}$: the numbers $1, 3, 6, 10, 15, \dots$, which count the dots in triangles of dots. | 5.3 |
Ch 6 · Triangles
| Term | Meaning | Used in |
|---|---|---|
| AA similarity criterion | If two angles of one triangle equal two angles of another, the triangles are similar (the third angles then match automatically because the angles of a triangle add up to $180^\circ$). | 6.3 7.2 8.1 |
| alternate angles | Angles on opposite sides of a transversal that cuts two parallel lines, lying between the lines. Alternate angles on parallel lines are equal. | 6.3 9.1 9.2 |
| Basic Proportionality Theorem | If a line parallel to one side of a triangle meets the other two sides at distinct points, it divides those sides in the same ratio: $DE\parallel BC\Rightarrow\dfrac{AD}{DB}=\dfrac{AE}{EC}$. | 6.2 6.3 |
| congruent figures | Figures of exactly the same shape and size — similar figures whose scale factor is $1$. | 6.1 6.2 6.3 8.2 |
| converse | The statement you get by swapping the "if" part and the "then" part of a theorem. A converse of a true theorem is not automatically true and needs its own proof. | 6.2 7.1 10.1 |
| converse of the Basic Proportionality Theorem | If a line meets two sides of a triangle at points that divide them in the same ratio, then the line is parallel to the third side: $\dfrac{AD}{DB}=\dfrac{AE}{EC}\Rightarrow DE\parallel BC$. | 6.2 |
| corresponding angles | Angles in matching positions where a transversal crosses two lines. If the two lines are parallel, corresponding angles are equal, and conversely. | 6.1 6.2 6.3 7.2 |
| corresponding sides | In similar triangles, sides that join matching vertices; they are opposite equal angles. In $\triangle ABC\sim\triangle PQR$, $AB$ corresponds to $PQ$. | 6.1 6.2 6.3 7.2 |
| corresponding vertices | The vertices that match in a similarity statement: in $\triangle ABC\sim\triangle PQR$ the pairs are $A\leftrightarrow P$, $B\leftrightarrow Q$, $C\leftrightarrow R$, in the order the letters are written. | 6.1 |
| enlargement | A similar copy of a figure that is larger than the original (scale factor $k\gt1$). A reduction has $0\lt k\lt1$. | 6.1 |
| included angle | The angle between two given sides of a triangle, at their common vertex. In $\triangle ABC$ the included angle of $AB$ and $AC$ is $\angle A$. | 6.3 |
| midpoint of a side | The point that divides a segment into two equal parts. The segment joining the midpoints of two sides of a triangle is parallel to the third side. | 6.2 6.3 7.1 7.2 |
| SAS similarity criterion | If one angle of a triangle equals one angle of another and the two sides including those angles are proportional, the triangles are similar. | |
| scale factor | The common ratio $k$ of corresponding sides of two similar figures. It multiplies every length (sides, perimeter) and leaves every angle unchanged. | 6.1 6.3 |
| similar figures | Figures with the same shape but not necessarily the same size. Two polygons are similar when their corresponding angles are equal and their corresponding sides are in the same ratio. | 6.1 |
| similar triangles | Triangles with equal corresponding angles and corresponding sides in the same ratio: $\triangle ABC\sim\triangle PQR$ means $\angle A=\angle P$, $\angle B=\angle Q$, $\angle C=\angle R$ and $\dfrac{AB}{PQ}=\dfrac{BC}{QR}=\dfrac{CA}{RP}$. | 6.1 6.3 7.2 8.1 |
| similarity criteria | Short tests (AA, SSS, SAS) that prove two triangles are similar without checking all three angles and all three ratios of sides. | 6.3 |
| similarity statement | A statement such as $\triangle ABC\sim\triangle PQR$ in which the order of the letters shows which vertices correspond. | 6.1 6.3 |
| SSS similarity criterion | If the three sides of one triangle are proportional to the three sides of another, $\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{CA}{FD}$, the triangles are similar. Compare shortest with shortest, longest with longest. | |
| transversal | A line that crosses two or more other lines. When it crosses two parallel lines, corresponding angles are equal. | 6.2 6.3 9.1 10.1 |
| trapezium | A quadrilateral with one pair of parallel sides (the bases). In a trapezium $ABCD$ with $AB\parallel DC$, the diagonals meet at $O$ with $\dfrac{AO}{OC}=\dfrac{BO}{OD}$. | 6.2 6.3 |
| vertically opposite angles | The equal angles formed opposite each other where two straight lines cross. They give an easy pair of equal angles when looking for similar triangles. | 6.3 |
Ch 7 · Coordinate Geometry
| Term | Meaning | Used in |
|---|---|---|
| abscissa | The $x$-coordinate of a point $(x, y)$: its signed distance from the $y$-axis. | 7.1 |
| Cartesian plane | A flat surface with two perpendicular number lines, the $x$-axis and the $y$-axis, crossing at the origin. Every point is described by an ordered pair $(x, y)$. | 7.1 |
| collinear points | Points that all lie on one straight line. For three points, the largest of the three distances equals the sum of the other two. | 7.1 |
| diagonals bisect each other | Two segments bisect each other when they have the same midpoint. This is true for the diagonals of every parallelogram, and is used to find a missing vertex. | 7.2 |
| distance formula | The length of the segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, from Pythagoras in a right triangle with horizontal and vertical legs. | 7.1 7.2 |
| equidistant | At the same distance from two (or more) given points or objects. $P$ is equidistant from $A$ and $B$ when $PA=PB$. | 7.1 7.2 10.2 |
| internal division | Dividing a segment $AB$ at a point $P$ lying between $A$ and $B$ so that $AP:PB=m_1:m_2$. | 7.2 |
| midpoint | The point halfway along a segment. For $A(x_1, y_1)$ and $B(x_2, y_2)$ it is $\left(\tfrac{x_1+x_2}{2}, \tfrac{y_1+y_2}{2}\right)$. | 6.2 6.3 7.1 7.2 |
| ordinate | The $y$-coordinate of a point $(x, y)$: its signed distance from the $x$-axis. | 7.1 |
| quadrant | One of the four regions into which the two axes divide the plane: I $(+,+)$, II $(-,+)$, III $(-,-)$, IV $(+,-)$. | 7.1 11.1 |
| ratio of division | The comparison $AP:PB$ of the two parts into which a point $P$ cuts a segment $AB$. It is often written $k:1$ when the exact parts are unknown. | 6.2 7.2 |
| rhombus | A quadrilateral with all four sides equal. If its diagonals are also equal, it is a square. | 6.1 7.1 10.2 |
| section formula | The formula for the point dividing the segment from $A(x_1, y_1)$ to $B(x_2, y_2)$ internally in the ratio $m_1:m_2$: $\left(\dfrac{m_1x_2+m_2x_1}{m_1+m_2},\ \dfrac{m_1y_2+m_2y_1}{m_1+m_2}\right)$. | 7.2 |
| trisection points | The two points that divide a segment into three equal parts. They divide it in the ratios $1:2$ and $2:1$. | 7.2 |
Ch 8 · Introduction to Trigonometry
| Term | Meaning | Used in |
|---|---|---|
| acute angle | An angle greater than $0^\circ$ and less than $90^\circ$. The two non-right angles of a right-angled triangle are acute. | 6.3 8.1 8.2 8.3 |
| adjacent side | The leg of a right-angled triangle that forms the acute angle you are looking at together with the hypotenuse. | 7.1 8.1 8.2 9.1 |
| altitude of a triangle | The perpendicular segment from a vertex to the opposite side (or to its extension). | 8.2 |
| cosine | For an acute angle $A$: $\cos A=\dfrac{\text{adjacent}}{\text{hypotenuse}}$. | 8.1 8.2 8.3 |
| counter-example | A single case in which a claimed formula fails. One counter-example is enough to show that a statement is not an identity. | 1.3 5.1 6.1 8.2 |
| equilateral triangle | A triangle with all three sides equal; each angle is $60^\circ$. Cutting it along an altitude gives a $30^\circ$–$60^\circ$–$90^\circ$ triangle. | 6.1 7.1 8.2 11.2 |
| exact value | A value written with fractions and roots, such as $\dfrac{\sqrt3}{2}$, instead of a rounded decimal such as $0.866$. | 3.2 7.1 8.2 9.1 |
| fundamental identity | The identity $\sin^2A+\cos^2A=1$. It is the Pythagoras theorem written with ratios, and the other two standard identities come from it by division. | 8.3 |
| hypotenuse | The side of a right-angled triangle opposite the right angle. It is the longest side. | 1.3 4.1 4.2 4.3 |
| isosceles right triangle | A right-angled triangle whose two legs are equal. Its acute angles are both $45^\circ$ and its sides are in the ratio $1:1:\sqrt2$. | 7.1 8.2 |
| left-hand side (LHS) | The expression to the left (LHS) or right (RHS) of the equals sign. To prove an identity, work on one side until it becomes the other. | 1.3 3.1 8.1 8.2 |
| not defined | A ratio is not defined when it needs a division by zero, for example $\tan90^\circ=\dfrac{1}{0}$ or $\operatorname{cosec}0^\circ$. | 2.1 8.2 |
| opposite side | The leg of a right-angled triangle that does not touch the acute angle you are looking at; the side that angle "faces". | 6.3 7.1 7.2 8.1 |
| reciprocal ratios | The three ratios $\operatorname{cosec}A=\dfrac1{\sin A}$, $\sec A=\dfrac1{\cos A}$ and $\cot A=\dfrac1{\tan A}$. | 8.1 8.3 |
| set square | A flat triangular drawing tool from a geometry box. The two usual ones have angles $45^\circ$–$45^\circ$–$90^\circ$ and $30^\circ$–$60^\circ$–$90^\circ$. | 6.2 8.1 8.2 10.1 |
| sine | For an acute angle $A$: $\sin A=\dfrac{\text{opposite}}{\text{hypotenuse}}$. | 8.1 8.2 |
| standard angles | The five angles $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$, whose trigonometric ratios can be written exactly (with fractions and square roots). | 8.2 8.3 9.1 |
| tangent ratio | For an acute angle $A$: $\tan A=\dfrac{\text{opposite}}{\text{adjacent}}=\dfrac{\sin A}{\cos A}$. (Not to be confused with a tangent line to a circle.) | 8.1 8.2 8.3 |
| trigonometric identity | An equation between trigonometric ratios that is true for every angle for which both sides are defined, for example $\sin^2A+\cos^2A=1$. | 8.3 |
| trigonometric ratio | The ratio of two sides of a right-angled triangle, seen from one of its acute angles. There are six: sine, cosine, tangent, cosecant, secant and cotangent. Each depends only on the angle. | 8.1 8.2 8.3 9.1 |
Ch 9 · Some Applications of Trigonometry
| Term | Meaning | Used in |
|---|---|---|
| angle of depression | The angle between the horizontal line through the observer's eye and the line of sight, when the object is below the horizontal and the observer looks down. It equals the angle of elevation of the observer as seen from the object. | 9.1 9.2 |
| angle of elevation | The angle between the horizontal line through the observer's eye and the line of sight, when the object is above the horizontal and the observer looks up. | 9.1 9.2 |
| baseline | A known, measured distance between two observation points; together with two angles it lets you find heights and distances you cannot measure directly. | 9.2 |
| clinometer | A simple instrument (or phone app) that measures the angle of elevation or depression of the line of sight. | 9.1 9.2 |
| eye height (of the observer) | The height of the observer's eyes above the ground; it must be added to the height found from the triangle when the observer is not at ground level. | 8.2 9.1 9.2 |
| flagstaff | A vertical pole fixed on top of a building or tower; its length is the difference between the elevations of its top and bottom seen from the same point. | 6.1 8.2 9.2 |
| heights and distances | The class of problems in which an unknown height or distance is found from measured angles of elevation or depression and one known length, using right-angled triangles. | 9.1 |
| horizontal line (level line) | The line through the observer's eye that is parallel to the ground. Angles of elevation and depression are always measured from it. | 9.1 |
| lighthouse | A tall tower near the sea, a favourite observation point in depression problems: the boats seen from its top have angles of depression equal to their angles of elevation seen from the boats. | 9.1 9.2 11.1 |
| line of sight | The straight line from the observer's eye to the object being looked at. | 9.1 9.2 10.1 10.2 |
| observation point | A place on the ground (or on a building) from which the angle to an object is measured. | 9.2 |
| same side and opposite sides (of a tower) | Two observation points are on the same side of a tower if the foot of the tower is not between them; on opposite sides if the foot lies between them. On the same side the distances subtract, on opposite sides they add. | 6.3 7.2 8.2 9.1 |
Ch 10 · Circles
| Term | Meaning | Used in |
|---|---|---|
| chord | A line segment joining two points of a circle. A diameter is the longest chord. The perpendicular from the centre to a chord bisects the chord. | 10.1 10.2 11.2 |
| chord of contact | The chord $AB$ joining the two points of contact of the tangents $PA$ and $PB$ from an external point $P$. The line $OP$ is its perpendicular bisector. | 10.2 |
| circumscribe | A polygon circumscribes a circle when every side of the polygon touches the circle. For a quadrilateral $ABCD$ this gives $AB+CD=BC+DA$. | 10.2 |
| concentric circles | Circles (or arcs) that have the same centre but different radii. | 10.1 11.1 |
| external point | A point lying outside a circle. Exactly two tangents can be drawn to the circle from an external point, and they have equal lengths. | 10.1 10.2 |
| incircle | The circle that lies inside a polygon and touches all its sides. For a triangle it is the circle touching all three sides; its centre is where the angle bisectors meet. | 10.2 |
| length of a tangent | The distance from an external point $P$ to the point of contact $A$ of a tangent from $P$. It equals $\sqrt{OP^2-r^2}$, where $O$ is the centre and $r$ the radius. | 10.1 10.2 |
| perpendicular distance | The distance from a point to a line measured along the perpendicular from the point to the line. It is the shortest distance from the point to the line. | 6.2 10.1 |
| point of contact | The single point where a tangent touches a circle. The radius to the point of contact is perpendicular to the tangent. | 10.1 10.2 |
| RHS congruence | Two right-angled triangles are congruent if the hypotenuse and one other side of one equal the hypotenuse and the corresponding side of the other. | 10.2 |
| secant | A line that cuts a circle at two distinct points. The part of a secant inside the circle is a chord. A secant at perpendicular distance $d$ from the centre satisfies $d\lt r$. | 8.1 8.3 10.1 |
| tangent to a circle | A line that meets a circle at exactly one point. It is perpendicular to the radius drawn to that point, and its distance from the centre equals the radius. | 10.1 10.2 |
| Term | Meaning | Used in |
|---|---|---|
| arc | The part of a circle between two points on it. The shorter one is the minor arc and the longer one the major arc. | 6.1 6.3 9.1 10.2 |
| central angle | The angle between the two radii that end at the ends of an arc. It decides what fraction of the circle the arc, and the sector, make up. | 10.2 11.1 11.2 |
| inscribed polygon | A polygon whose corners all lie on a circle. An inscribed regular polygon with $n$ sides leaves $n$ equal segments between itself and the circle. | |
| major segment | The larger of the two segments made by a chord that is not a diameter; it contains the centre of the circle. Its area is the circle minus the minor segment. | 11.2 |
| minor segment | The smaller of the two segments made by a chord that is not a diameter; it lies on the side of the chord away from the centre. Its area is sector minus triangle. | 11.2 |
| perimeter of a sector | The total length of the boundary of a sector: the arc length plus the two radii, $P=l+2r$. | 11.1 |
| quadrant of a circle | A sector whose central angle is $90^\circ$, that is, one quarter of a circle. | 11.1 |
| regular hexagon | A six-sided figure with all sides and angles equal. Inside a circle its side equals the radius, because each of its six central triangles is equilateral. | 6.1 11.2 |
| sector of a circle | The region of a circle enclosed by two radii and the arc between them, like a slice of pizza. Its area is the fraction $\dfrac{\theta}{360}$ of the whole circle. | 11.1 11.2 12.1 14.1 |
| sector of a ring | The region between two concentric arcs with the same central angle, bounded also by two straight edges. Its area is the difference of two sector areas. | 11.1 |
| segment of a circle | The region of a circle cut off by a chord, bounded by that chord and the arc on one side of it. Not to be confused with a sector, which is bounded by two radii. | 11.2 |
| semicircle | A sector (or an arc) with central angle $180^\circ$: exactly half of a circle. | 11.1 11.2 |
Ch 12 · Surface Areas and Volumes
| Term | Meaning | Used in |
|---|---|---|
| capacity | The volume of liquid or grain a hollow container can hold, often given in litres or millilitres. | 12.2 |
| cavity | A space cut or drilled out of a solid, such as a conical hole in a wooden cylinder. Its volume is subtracted from the volume of the solid. | 12.1 12.2 |
| combination of solids | A solid made by joining two solids face to face, such as a cone on a hemisphere or a cylinder with a hemisphere at each end. | 12.1 12.2 |
| contact area | The part of the surfaces of two joined solids that touch. It is hidden inside, so it is subtracted twice from the sum of their total surface areas. | 12.1 |
| cubic unit | A unit of volume, the volume of a cube of edge one unit (for example one cubic centimetre, 1 cm³). | 12.2 |
| curved surface area (CSA) | The area of the curved part of a solid only (the wall of a cylinder, the sloping surface of a cone, the dome of a hemisphere), without its flat faces. Written CSA. | 12.1 |
| density | The mass of a material per unit volume, in g/cm³. Mass $=$ volume $\times$ density. | 12.2 |
| hemisphere | Half of a sphere, cut by a plane through its centre. It has one curved surface of area $2\pi r^2$ and one flat circular face of area $\pi r^2$. | 12.1 12.2 |
| litre | A unit of capacity: $1\ \text{L}=1000\ \text{cm}^3$, so $1\ \text{m}^3=1000\ \text{L}$. | 2.1 3.1 3.3 12.2 |
| right circular cone | A cone with a circular base whose axis (the line from the apex to the centre of the base) is at right angles to the base. | 12.1 |
| right circular cylinder | A cylinder with two parallel, equal circular faces whose axis is at right angles to both, like a tin or a water pipe. | 12.1 |
| scooped-out solid | A solid from which a piece (such as a hemisphere or a cone) has been removed. The opening disappears from the outside and the inner surface of the hollow becomes new outside. | 12.1 12.2 |
| slant height | For a cone, the length from the apex to a point on the edge of the base. It is found from $l=\sqrt{r^2+h^2}$ and is used in the curved surface area $\pi rl$. | 12.1 12.2 |
| total surface area (TSA) | The area of the whole outer boundary of a solid: the curved surface plus every flat face that lies on the outside. Written TSA. | 12.1 12.2 |
| volume of a solid | The amount of space a solid occupies, measured in cubic units such as cm³ or m³. | 12.2 |
Ch 13 · Statistics
| Term | Meaning | Used in |
|---|---|---|
| assumed mean | Any convenient number $a$ (usually a class mark near the middle) from which deviations are measured to shorten the arithmetic. The final mean does not depend on the choice of $a$. | 13.1 |
| class mark | The midpoint of a class interval: $\dfrac{\text{upper limit}+\text{lower limit}}{2}$. In calculations every observation in the class is treated as if it equalled the class mark. | 13.1 13.2 |
| class size | The width of a class interval, found as upper limit $-$ lower limit. It is written $h$. | 13.1 13.2 13.3 |
| continuous classes | Classes that touch, such as $10$–$20$, $20$–$30$, with no gap between the upper limit of one and the lower limit of the next. Classes written $10$–$19$, $20$–$29$ are made continuous by subtracting $0.5$ from every lower limit and adding $0.5$ to every upper limit. | 13.2 13.3 |
| cumulative frequency | The running total of frequencies: the number of observations in a class and in all the classes before it. The cumulative frequency of the last class is the total $n$. | 13.3 |
| deviation | The difference $d_i=x_i-a$ between a class mark and the assumed mean. It is negative for classes below $a$. | 8.3 13.1 |
| empirical relation | The observed rule $3\,\text{Median}=\text{Mode}+2\,\text{Mean}$, which holds approximately for moderately skewed distributions and is used as a check or to estimate one average from the other two. | 13.3 |
| frequency | The number of observations that fall in a class (or that take a given value). It is written $f_i$. | 13.1 13.2 13.3 |
| grouped data | Data sorted into class intervals (such as $0$–$10$, $10$–$20$) with a count of how many observations fall in each class. The individual values are not shown. | 13.1 13.2 13.3 |
| mean of grouped data | The average $\bar x=\dfrac{\sum f_ix_i}{\sum f_i}$ found by weighting each class mark $x_i$ by its frequency $f_i$. | 13.1 |
| median | The middle value when the observations are arranged in order: half of the observations are at most the median and half are at least the median. | 6.3 13.3 |
| median class | The first class whose cumulative frequency is at least $\dfrac n2$. The median of the grouped data lies in this class. | 13.3 |
| modal class | The class of a grouped frequency distribution that has the highest frequency. The mode of the grouped data lies in this class. | 13.2 |
| mode | The value that occurs most often in a set of observations. For grouped data it is estimated inside the modal class with a formula. | 8.1 13.2 13.3 |
| skewed distribution | A distribution whose histogram has a long tail on one side, so that the mean, median and mode are different. A long right tail pulls the mean above the median. | 13.3 |
| step-deviation method | A shortcut in which the deviations are divided by the class size, $u_i=\dfrac{x_i-a}{h}$, giving $\bar x=a+h\dfrac{\sum f_iu_i}{\sum f_i}$. | 13.1 |
Ch 14 · Probability
| Term | Meaning | Used in |
|---|---|---|
| at least one | "One or more". The probability of "at least one" is usually found from the complement "none": $P(\text{at least one})=1-P(\text{none})$. | 1.1 3.1 4.2 14.2 |
| complementary event | The event "$E$ does not happen", written $\overline{E}$ and read "not $E$". Its probability is $1-P(E)$. | 14.1 |
| doublet | A result of throwing two dice in which both show the same number, such as $(4,4)$. There are $6$ doublets among the $36$ outcomes. | 14.2 |
| equally likely outcomes | Outcomes none of which is more likely to occur than another, as when a fair coin is tossed or a card is drawn from a well-shuffled pack. The counting formula for probability needs them. | 4.3 14.1 14.2 |
| event | A collection of outcomes of an experiment, such as "an even number" when a die is thrown. | 1.2 14.1 14.2 |
| face card | A jack, queen or king. A pack has $12$ face cards, $3$ in each suit; aces are not face cards. | 14.2 |
| fair coin or die | A coin or die that is tossed or thrown so that all its faces are equally likely, so the counting formula for probability applies. | 14.1 14.2 |
| favourable outcome | An outcome that makes a given event happen. For the event "a prime number" on a die, the favourable outcomes are $2,3,5$. | 14.1 14.2 |
| impossible event | An event with no favourable outcome; it can never happen and its probability is $0$. | 14.1 14.2 |
| ordered pair of outcomes | A result written as $(a,b)$ with $a$ from the first die or coin and $b$ from the second, so that $(2,5)$ and $(5,2)$ are different outcomes. | 7.1 |
| outcome | One possible result of a random experiment. Tossing a die has six outcomes: $1,2,3,4,5,6$. | 14.1 14.2 |
| random experiment | An activity, such as tossing a coin or drawing a ball from a bag, whose result cannot be predicted with certainty although all the possible results are known. | 14.1 |
| sample space | The list of all possible outcomes of a random experiment, for example $\{H,T\}$ for one coin. | 14.1 14.2 |
| spinner | A wheel with a pointer that stops in one of several sectors. If the sectors are equal, each is equally likely, and $P(E)$ is the number of favourable sectors divided by the total. | 14.1 14.2 |
| standard pack of cards | A set of $52$ playing cards in $4$ suits of $13$ cards, with $26$ red and $26$ black cards; a card drawn from a well-shuffled pack is equally likely to be any of the $52$. | 14.2 |
| suit | One of the four groups of $13$ cards in a pack: spades and clubs (black), hearts and diamonds (red). | 5.3 14.2 |
| sure event | An event that every outcome favours; it always happens and its probability is $1$. | 14.1 |
| theoretical probability | For equally likely outcomes, $P(E)=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$, found by reasoning rather than by repeating the experiment. | 14.1 |