Your coordinate geometry toolkit

Three formulas describe where points sit relative to each other on the Cartesian plane, and everything else in this chapter builds on them. None of the three appear on the exam formula list, so they are worth committing to memory in a form you can write out without hesitation.

$$d = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$$

$$\left(\dfrac{x_{1}+x_{2}}{2},\ \dfrac{y_{1}+y_{2}}{2}\right)$$

$$\left(\dfrac{nx_{1}+mx_{2}}{m+n},\ \dfrac{ny_{1}+my_{2}}{m+n}\right)$$

The section formula is the general case; the midpoint formula is just the section formula with $m=n$, so a point dividing a segment equally lands exactly halfway. Getting the labelling of $(x_{1},y_{1})$ and $(x_{2},y_{2})$ consistent with which end of the ratio $m:n$ is closer to which point matters more than the arithmetic, a swapped label is the most common way this formula goes wrong.

Once a line's gradient is known, comparing it with another line's gradient answers most "are these lines parallel or perpendicular" questions without needing to sketch anything.

• Parallel lines: $m_{1}=m_{2}$, the gradients are identical.
• Perpendicular lines: $m_{1}\times m_{2}=-1$, the gradients multiply to give $-1$, which is the same as saying each gradient is the negative reciprocal of the other.

The perpendicular condition is the one worth double-checking under exam pressure, because the negative reciprocal step is where slips happen: the gradient of the perpendicular to a line of gradient $\dfrac{2}{3}$ is $-\dfrac{3}{2}$, not $\dfrac{3}{2}$ and not $-\dfrac{2}{3}$.

The area formula lets you find the area of a triangle or polygon directly from its vertices' coordinates, without measuring a base or height on a diagram.

$$\text{Area} = \tfrac{1}{2}\left|x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_2)\right|$$

The pattern is easier to remember as a cross-multiplication down two columns: list the coordinates in order, going around the shape, repeat the first point at the end, then sum the products going one diagonal direction and subtract the sum going the other. For a polygon with more than three vertices, the same pattern extends, list every vertex in order around the shape and continue the same diagonal sums.

A locus is the path traced out by a point that moves according to a fixed condition. Add Math focuses on two conditions in particular, and both convert into an equation the same way: let the moving point be $(x,y)$, write the condition algebraically using the distance formula, then simplify.

• Distance from a fixed point is constant, this always produces a circle. A point $(x,y)$ at constant distance $r$ from a fixed point $(a,b)$ satisfies $(x-a)^{2}+(y-b)^{2}=r^{2}$.
• Ratio of distances from two fixed points is constant, this traces a circle too (unless the ratio is $1:1$, which gives the perpendicular bisector, a straight line instead).

Squaring both sides of the distance condition before simplifying avoids working with square roots throughout the algebra, and is the standard first move for both locus types.

Coordinate geometry questions are rarely conceptually difficult once the five core formulas are secure, most lost marks trace back to a small handful of avoidable slips.

• Mixing up which point is $(x_1,y_1)$ and which is $(x_2,y_2)$ partway through a formula, especially in the section formula where the ratio order matters.
• Forgetting the negative reciprocal when finding a perpendicular gradient, and using the plain reciprocal instead.
• Dropping the absolute value bars in the area formula and reporting a negative area.
• Leaving a locus equation unsimplified, or in a form that does not match what the question specifically asks for (equation of a circle vs. equation of a straight line).

Coordinate geometry questions often stack two or three of these five formulas in a single problem, finding a midpoint, then a gradient, then an equation, for instance, and a small error early on quietly propagates through every later step. Watching a student's full working line by line makes it possible to catch exactly where that first error happened, rather than only seeing that the final answer is wrong.

Our teachers are experienced; lessons run online and are taught in English, while SPM papers themselves are set bilingually in Bahasa Melayu and English.

If coordinate geometry, or keeping five memorised formulas straight under exam pressure, is a recurring difficulty, a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience) is a reasonable way to see how targeted, one-to-one practice helps. We will not promise a particular grade, but we can help this toolkit stop feeling like five separate things to recall and start feeling like one connected way of reading a Cartesian plane.