📍Coordinate Geometry

Master Pytha surveys Pythagoras Pass with pegs and a chain, and he never measures anything twice: once two pegs have coordinates, everything about the segment between them can be calculated. Three formulas do the whole job, and none of them is worth memorising until you have seen where it comes from.

The midpoint is the average of the two ends, taken one coordinate at a time:

M = ((x1 + x2) / 2, (y1 + y2) / 2)

Halfway across and halfway up is halfway along — that is the whole idea. Notice that it ADDS the coordinates and then halves. Subtracting and halving gives you half the journey, which is a direction, not a place.

The distance is the Pythagorean theorem wearing coordinates. Draw a right triangle under the segment with one leg running across and one running up, and the segment itself is the hypotenuse:

d = sqrt((x2 - x1)² + (y2 - y1)²)

The two subtractions can come out negative, and it does not matter in the least, because squaring makes both of them positive. What does matter is the square root at the end: stop one step early and you have reported d² instead of d.

The slope measures the steepness of the segment as rise over run:

m = (y2 - y1) / (x2 - x1)

The change in y sits on top. If the two pegs happen to have the same x-coordinate the run is 0, and since nothing may be divided by 0, that segment has no slope number at all — it is vertical, and you say so in words.

Two segments are parallel when they have exactly the same slope. That is the easy test, and it stays true however long the segments are and wherever they sit on the grid — slope is about direction, not position.

Two segments are perpendicular when their slopes multiply to -1. Another way to say the same thing: one slope is the other one turned upside down AND given the opposite sign. That is why 2/3 and -3/2 are perpendicular, while 2/3 and 3/2 are not, and neither are 2/3 and -2/3.

parallel: m1 = m2 perpendicular: (m1)(m2) = -1

Both tests need the two slopes worked out the same way round. Take the points in one order for the first fraction and the other order for the second, and you flip the sign of one slope without meaning to — which turns a parallel pair into a pair that looks like nothing at all.

Vertical segments are the one case where multiplying is no help, because a vertical segment has no slope number to multiply. Handle it by looking instead:

• ◆vertical with vertical: parallel.
• ◆vertical with horizontal (slope 0): perpendicular.
• ◆vertical with anything slanted: neither.

A slope of 0 is a perfectly good number, by the way, and it is not the same thing as having no slope. A horizontal segment has slope 0; a vertical one has no slope at all.

A midpoint splits a segment 1 : 1. Any other split is asked for as a ratio, and the trap is built into the way ratios are written.

If P divides the directed segment from A to B so that AP : PB = 2 : 3, then AB has been cut into 2 + 3 = 5 equal parts and P sits after two of them. So P is 2/5 of the way from A to B, not 2/3. The parts have to be added before anything is divided.

The word directed tells you which end to count from. "From A to B" means start at A, and the first number of the ratio is the piece nearest A. Read the ratio from the wrong end and you land at the mirror-image point, which is a different place unless the ratio happens to be 1 : 1.

The method is then three short steps:

change from A to B: (x2 - x1, y2 - y1) take your share: (a / (a + b)) of each change add the share to A: P = (x1 + share across, y1 + share up)

For example, from A(1, 2) to B(11, 7) in the ratio 3 : 2, the whole change is (10, 5), the share is 3/5 of each, which is (6, 3), and P = (1 + 6, 2 + 3) = (7, 5). The last step is the one people drop: the share tells you how far to travel, and travelling has to start from A.

Give a figure coordinates and its name stops being a matter of opinion. Two tools settle everything.

Squared lengths decide whether sides are equal. You never need the square roots for this: if AB² = BC² then AB = BC, because both lengths are positive. Keeping the squares makes the arithmetic exact and the comparison instant.

Slopes decide whether sides are parallel or perpendicular, exactly as in the section above.

For a triangle, ask two questions. Are any two sides equal? Is any corner a right angle? A corner is right when the two smaller squared sides add to the largest one — the converse of the Pythagorean theorem — or when the slopes of the two sides meeting there multiply to -1. The two answers together give the description, which is why the options are written as pairs of facts: "right-angled and isosceles", "isosceles, but not right-angled", "right-angled, but not isosceles", "scalene, with no right angle". One and only one of them can be true.

Equilateral is never an option on a grid. No triangle with whole-number corners can have all three sides equal, so if you find yourself choosing it, two of the squared sides will disagree on a second look.

For a quadrilateral with its corners listed in order, work through the family in order of fussiness:

• ◆both pairs of opposite sides parallel: a parallelogram at least. Then equal adjacent sides make it a rhombus, a right angle makes it a rectangle, and both together make it a square.
• ◆exactly one pair of opposite sides parallel: a trapezoid.
• ◆no parallel sides but two pairs of equal sides sitting next to each other: a kite.

Every square is also a rectangle, a rhombus and a parallelogram, so the question always asks for the most specific name — the smallest family the shape belongs to. Stopping at "parallelogram" when the corners are square is the commonest way to lose a right answer.

Perimeter is the sum of the side lengths, one distance formula per side. Sides that run along a grid line need no formula at all: just subtract the coordinate that changes. Round only once, at the very end, after every side has been added.

Area is easiest with the shoelace formula, which needs the corners listed in order round the shape and no square roots at all. Pair each x with the NEXT y and add those products; pair each y with the next x and add those; subtract, take the size of the result, and halve it. For a triangle A(1, 1), B(5, 2), C(2, 6):

one way: (1)(2) + (5)(6) + (2)(1) = 34 other way: (1)(5) + (2)(2) + (6)(1) = 15 area = |34 - 15| / 2 = 19 / 2 = 9.5

The halving at the end is the step that goes missing most often. A negative difference simply means you walked round the shape the other way, so drop the minus sign.

Now the conventions this skill uses when it asks you for an answer.

• ◆A point is typed as an ordered pair. "(3, -2)", "3, -2" and "x = 3, y = -2" are all accepted. When a midpoint lands between grid lines, write the half as a decimal — "2.5" — or as a fraction, "5/2"; both pass, and the prompt says decimals are fine.
• ◆A length or a perimeter is a single number. When it comes out whole, give the whole number. When it does not, the prompt says "round to 1 decimal place", so sqrt(97) is answered 9.8.
• ◆An area is a single number too, and it may end in .5. A lattice triangle very often has an area like 9.5, so decimals are welcome and 19/2 is accepted as well.
• ◆Classifications are multiple choice, and the options are written so that exactly one of them is true: the most specific name for a quadrilateral, and a description of a triangle that states both facts about it. Here "trapezoid" means exactly one pair of parallel sides, so a parallelogram is never called one.