Coordinate Geometry

• The one-line summary: find the gradient, substitute a point, simplify, then check the point still fits. That single routine answers subtopics 3.5, 3.6 and 3.7 — only the source of the gradient changes.
• Parallel → same. Perpendicular → flip and negate. If you can say that sentence under pressure, and you remember that vertical lines are the exception to almost everything, four of the six danger zones above stop being dangerous. The remaining two — subtraction order, and the bisector’s second condition — are caught by checking your answer rather than by remembering a rule.
• The intercept shortcut. For any equation in the form \(ax+by=c\), setting \(x=0\) gives the \(y\)-intercept and setting \(y=0\) gives the \(x\)-intercept. Two substitutions, two points, one ruled line — often faster than building a table.
• Five checks that catch all six. Does the plotted point match its ordered pair? Did I subtract in the same order top and bottom? Is this line vertical? Did the question say parallel or perpendicular? Does my final equation contain the point it is supposed to pass through?
• Parallel → same gradient. Perpendicular → flip it and negate it. Vertical lines break every rule and have no gradient at all.
• Ten minutes today, fifteen on day 3, twenty on day 10, ten on day 30. Fifty-five minutes in total — less than one revision session, spread so that it survives to the examination.

• 3.1 Plot and read a point as an ordered pair \((x,\,y)\), state which quadrant it lies in from the signs of its coordinates, and recognise that a point on the \(x\)-axis has \(y=0\) and a point on the \(y\)-axis has \(x=0\).
• 3.2 Draw the graph of a linear equation from a table of values or from its two intercepts, rearranging into \(y=mx+c\) when that is quicker, and label the axes, the scale and the line.
• 3.2 Recognise \(x=k\) as a vertical line with undefined gradient and \(y=k\) as a horizontal line with gradient \(0\).
• 3.3 Find a gradient from two coordinates or by drawing a gradient triangle on a graph, subtracting in a consistent order and simplifying the fraction.
• 3.3 Interpret the sign and size of a gradient, and say when a gradient is zero and when it is undefined.
• 3.4 Calculate the length of a line segment using \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\), leaving a surd exact when an exact answer is required.
• 3.4 Calculate the midpoint of a line segment as the average of the two \(x\)-coordinates and the average of the two \(y\)-coordinates.
• 3.5 Obtain the equation of a line from two points, from one point and a gradient, or from a drawn graph, and give it in the form the question asks for.
• 3.5 Rearrange between \(y=mx+c\), \(y-y_1=m(x-x_1)\) and \(ax+by=c\), and read the gradient and intercept off any of them.
• 3.6 Find the equation of the line through a given point parallel to a given line, and explain why equal gradients with equal intercepts describe one line rather than two.
• 3.7 Use \(m_1m_2=-1\) to find the gradient of a perpendicular line by taking the negative reciprocal, and handle the horizontal/vertical exception.
• 3.7 Construct the equation of the perpendicular bisector of a line segment, checking both that it passes through the midpoint and that it is perpendicular.

Where this shows up. Every digital map, every screen, every CAD drawing and every plotted data set is a coordinate plane. A phone locating a shop is comparing coordinate pairs; a graph of cost against time is reading a gradient as a rate. The algebra you learn here is the reason a computer can decide whether two drawn walls meet at a right angle without ever using a protractor.

Coordinates

An ordered pair (x, y) fixing the position of a point in a plane, measured against two perpendicular number lines: the x-coordinate gives the displacement from the origin parallel to the horizontal x-axis and is always written first, and the y-coordinate gives the displacement parallel to the vertical y-axis and is always written second. The point (0, 0) where the axes cross is the origin.

Drawing Linear Graphs

The process of converting a linear equation into an accurately ruled straight line on a coordinate grid, either by tabulating at least two correct points and plotting them to within half a small square, or by finding the two axis intercepts. A linear equation is one in which x and y appear only to the first power, so its graph is always a straight line; the vertical line x = k is the one case that cannot be written in the form y = mx + c.

Parallel Lines

Two distinct straight lines in the same plane that never meet, however far they are extended. For non-vertical lines this happens exactly when their gradients are equal, so a line parallel to y = mx + c has the same m but a different c. Two vertical lines x = a and x = b are also parallel whenever a and b differ. Equal gradients together with equal intercepts do not describe two parallel lines: they describe the same line written twice.

Gradient of Linear Graphs

A single number measuring the steepness and direction of a straight line, equal to the change in y divided by the corresponding change in x between any two points on the line. It is computed as (y2 minus y1) divided by (x2 minus x1) with the same subtraction order used in numerator and denominator. A positive gradient rises from left to right, a negative gradient falls, a horizontal line has gradient zero, and a vertical line has an undefined gradient because its change in x is zero.

Length and Midpoint

The length of a line segment joining two points is the square root of the sum of the squares of the horizontal and vertical changes between them, which is Pythagoras applied to the right-angled triangle whose hypotenuse is the segment; it is always non-negative. The midpoint of the segment is the point whose coordinates are the mean of the two x-coordinates and the mean of the two y-coordinates, and it lies exactly halfway along the segment.

Perpendicular Lines

Two straight lines that cross at an angle of 90 degrees. When both have a defined gradient, the product of their gradients is minus one, so each gradient is the negative reciprocal of the other: the fraction is inverted and its sign is changed. The one perpendicular pair this rule cannot describe is a horizontal line, of gradient zero, meeting a vertical line, whose gradient is undefined. The perpendicular bisector of a line segment is the line perpendicular to it that also passes through its midpoint.

Equations of Linear Graphs

An algebraic relationship satisfied by the coordinates of every point on a straight line and by no other point. The same line can be written in several equivalent forms: y = mx + c shows the gradient m and the y-intercept c directly, y - y1 = m(x - x1) is built from one known point and the gradient, ax + by = c clears fractions and is the usual integer form, and x = k describes a vertical line, which is the only straight line that cannot be written as y = mx + c.

• “Every straight line can be written as \(y=mx+c\).” Not true A vertical line cannot. \(x=4\) has no \(m\) and no \(c\), because its gradient is undefined. Any argument that starts “let the line be \(y=mx+c\)” silently assumes the line is not vertical.
• “A vertical line has gradient zero.” Reversed A horizontal line \(y=k\) has gradient \(0\). A vertical line \(x=k\) has gradient undefined, because the change in \(x\) is \(0\) and you cannot divide by \(0\).
• “It doesn't matter which point I call the first one.” Half true It genuinely does not matter — provided you use the same order in the numerator and the denominator. \(\dfrac{y_2-y_1}{x_2-x_1}\) and \(\dfrac{y_1-y_2}{x_1-x_2}\) are equal. Mixing them gives you the right size with the wrong sign.
• “Parallel means the gradients are negative reciprocals.” Swapped Parallel lines have equal gradients. Perpendicular lines have negative reciprocal gradients. The swap is a costly one, because it produces a tidy-looking answer that is nevertheless a completely different line.
• “For a perpendicular, just change the sign of the gradient.” Incomplete You must take the negative reciprocal: turn the fraction upside down and change the sign. From \(\dfrac{3}{4}\) you get \(-\dfrac{4}{3}\), not \(-\dfrac{3}{4}\). Check with \(m_1m_2=-1\) every time.
• “A perpendicular bisector just has to be perpendicular.” Half a line It must satisfy both conditions: perpendicular to the segment and passing through its midpoint. An answer that meets only one of the two is not the perpendicular bisector, however correct the gradient looks.

• Working matters as much as the answer. The syllabus is explicit about this: candidates must show all necessary working in the spaces provided, and where a question asks for working, a correct final answer on its own cannot earn full marks. Write \(m=\dfrac{y_2-y_1}{x_2-x_1}\) with the numbers in it, then the arithmetic, then the equation.
• Accuracy conventions that apply on both papers. A plotted point must be within half of the smallest grid square of its true position. A straight line must be ruled, and drawn across the whole domain the question gives. An equation should be fully simplified unless the question says otherwise: \(y=2x-5\), not \(y=\dfrac{4x-10}{2}\). When a question specifies a form — “in the form \(ax+by=c\) where \(a\), \(b\) and \(c\) are integers” — the answer has to be in that form. A correct line written in a form the question did not ask for has not answered the question. Read the form before you start. Answers are expected in their simplest form unless the question says otherwise, and where a question asks for an exact value, a surd or a fraction is what it wants. How accurately you may read a value off a graph is set by the scale of that graph. On a grid where one square is two units, quoting an answer to two decimal places is not justified.
• What the calculator will not do for you. It will not write your method, and the method is what you have to communicate in the examination. Use it the way you would use an answer key: calculate first, compare second, and when the two disagree, find which line of your working went wrong rather than copying the result.
• Match the requested form exactly. “In the form \(ax+by=c\) where \(a\), \(b\) and \(c\) are integers” means no fractions and no decimals anywhere. If you reach \(y=\tfrac{2}{3}x-\tfrac{1}{6}\), multiply through by \(6\) to get \(6y=4x-1\), then rearrange to \(4x-6y=1\). A correct line written in a form the question did not ask for has not yet answered the question, and converting takes one line.
• The habit these three share. In every one of them, a later part reuses a number from an earlier part instead of recomputing it. That is not just faster — it is the structure the question was written with, and following it keeps your working consistent. If part (c) makes you recalculate something you already found in part (a), read the question again.
• Marking yourself honestly. An equation that is algebraically equivalent to the printed answer is correct: \(3x+2y=14\) and \(y=-\tfrac{3}{2}x+7\) are the same line. An equation that merely looks similar is not. The test is not whether it resembles the answer, but whether the given point satisfies it and the gradient matches — apply that test rather than comparing shapes.
• After marking. Sort your errors into two piles: arithmetic slips, and method errors. Slips are fixed by writing more slowly and checking. Method errors — using the wrong rule, substituting the wrong point, misreading a form — are fixed by going back to the section that owns them. The two need completely different responses, and treating a method error as “a silly mistake” guarantees you make it again.
• Interleave rather than block. Once Topic 3 is solid, stop practising it on its own. Mix coordinate-geometry questions in with algebra and mensuration questions, so that you have to recognise which method a question wants before you can apply it. Recognition is the skill an examination actually tests, and a page of questions all from the same topic never trains it.