Coordinate Geometry · Class 10 Mathematics: quick revision, 49 bites

The distance formula

Pythagoras on the grid

Takeaway. Across squared plus up squared, then root.

The right triangle

Where it comes from

Draw a horizontal line from one point and a vertical line from the other.

Takeaway. The formula is Pythagoras, nothing more.

A(1, 1) to B(5, 4)

The oldest triangle

Takeaway. A 3-4-5 triangle, drawn on axes.

The order does not matter

Either way round

$(x_2 - x_1)^2 = (x_1 - x_2)^2$, because squaring destroys the sign.

Takeaway. Choose whichever order is easier to compute.

Negative coordinates

Minus a minus

For $A(-5, 7)$ and $B(-1, 3)$: $-1 - (-5) = 4$.

Takeaway. Get the difference right first.

Distance from the origin

The shortcut you will use most

Takeaway. P(3, 4) is 5 units from O.

Surds

Keep it exact

$\sqrt8 = 2\sqrt2$, not 2.83.

Takeaway. An exact answer is the expected one.

Compare the squares

Skip the roots

If $AB^2 = AC^2$ then $AB = AC$, because distances are positive.

Takeaway. This removes most of the arithmetic from hard questions.

From three points

What kind of triangle?

Takeaway. All three tests work on squares.

A(1, 7), B(4, 2), C(−1, −1)

Both at once

$AB^2 = 34$, $BC^2 = 34$, $CA^2 = 68$.

Takeaway. Three squares, no roots.

Finding a missing coordinate

A distance is given

Takeaway. Two answers are normal here.

Don't drop a root

Two points, not one

If $(3 - x)^2 = 9$ then $x = 0$ or $x = 6$.

Takeaway. Squaring always offers two roots.

Set the squares equal

Equidistant points

For a point on the x-axis, write it as $(x, 0)$ and set $PA^2 = PB^2$.

Takeaway. Much easier than it looks.

How to set out

Show the two gaps

Write the horizontal difference and the vertical difference separately before squaring.

Takeaway. Quote the formula first.

Distance is a length

Never negative

Always take the positive square root; a distance of $-5$ means nothing.

Takeaway. The $\pm$ belongs in coordinate equations, not in distances.

Where it is used

The same idea on a map

Grid references on a map and GPS coordinates both turn into distances by exactly this formula.

Takeaway. In three dimensions it simply gains a z term.

The section formula

A weighted average of the ends

Takeaway. Note which weight sits on which point.

The error that costs most

The weights are CROSSED

In $m : n$ from A, it is $m$ that multiplies $x_2$ — B's coordinate.

Takeaway. A big m means far from A, so B gets the weight.

The reason behind it

Why crossed?

A large $m$ puts the point FAR from A and CLOSE to B.

Takeaway. It is a balance, not a coincidence.

(−1, 7) and (4, −3) in 2 : 3

A worked division

Takeaway. Cross the weights, then divide by m + n.

2 : 3 is not 3 : 2

Order matters

$2 : 3$ from A is nearer A; $3 : 2$ from A is nearer B.

Takeaway. Fix A and B before substituting.

The midpoint formula

The simplest case

Takeaway. Equal weights give an ordinary average.

Midpoint is section at 1 : 1

Not a separate rule

There is only one formula to learn; the midpoint is what it becomes when the weights are equal.

Takeaway. That is why it looks like a mean.

(1, 2) and (7, 8)

A midpoint

$\left(\frac{1+7}{2}, \frac{2+8}{2}\right) = (4, 5)$.

Takeaway. Average each coordinate.

Trisection

Two points, not one

Takeaway. Never 1 : 3.

The trisection trap

One part out of three

A trisection point is one part along out of THREE, so the ratio is $1 : 2$.

Takeaway. Count the parts the ratio makes, not the pieces.

Let the ratio be k : 1

One unknown, not two

Every ratio $m : n$ can be written $k : 1$ with $k = \frac mn$.

Takeaway. This is the standard move for 'find the ratio' questions.

(−1, 6) on (−3, 10) to (6, −8)

Finding a ratio

$\frac{6k - 3}{k + 1} = -1$ gives $7k = 2$, so $k = \frac27$.

Takeaway. Then check with the y-coordinate.

Which coordinate to zero

Crossing an axis

Takeaway. The x-axis is where y is zero — say it aloud.

A(1, −5) to B(−4, 5)

An axis crossing

$\frac{5k - 5}{k + 1} = 0$ gives $k = 1$.

Takeaway. A ratio of 1 : 1 means the midpoint.

One line, most errors caught

Check with the other coordinate

Find the ratio from x, then substitute it back and confirm the y-coordinate.

Takeaway. Always verify.

What it means

A negative k

A negative k puts the point outside the segment, on the extension of the line.

Takeaway. At this level, re-check the arithmetic first.

Where it is used

Milestones and flags

Placing a flag halfway between two markers, or dividing a road into three equal stretches, is exactly this calculation.

Takeaway. The centroid of a triangle divides each median 2 : 1.

What this topic is

Three questions, two tools

Collinearity, classifying a quadrilateral, and finding a missing point.

Takeaway. Lengths → distance. Ratios → section.

The collinearity test

Short plus short equals long

Takeaway. No room left for a bend.

The collinearity trap

Do NOT square here

For collinear points $AB^2 = 10$, $BC^2 = 40$, $AC^2 = 90$ — and $10 + 40 \ne 90$.

Takeaway. The one place squares are the wrong tool.

(−2, 3), (1, 2), (7, 0)

A collinear trio

$AB = \sqrt{10}$, $BC = 2\sqrt{10}$, $AC = 3\sqrt{10}$.

Takeaway. Write each as a multiple of the same surd.

Classifying a quadrilateral

Six lengths, then decide

Takeaway. Squares are fine here — you are only comparing.

Sides and diagonals

The whole classification

Takeaway. Learn this table and the topic is half done.

Why diagonals matter

Four equal sides are not enough

Push a square sideways: the sides keep their lengths but it becomes a rhombus.

Takeaway. Always compute both diagonals.

Two worked cases

Square or rhombus?

$(1,1),(5,1),(5,5),(1,5)$: sides all 16, diagonals both 32 → SQUARE.

Takeaway. Same sides, different diagonals.

Square, rectangle or parallelogram?

Name the most specific

A square IS a rectangle and IS a rhombus — all the names are true.

Takeaway. Most specific wins.

Finding a missing vertex

Diagonals bisect each other

In a parallelogram both diagonals share one midpoint.

Takeaway. One property, a whole family of questions.

A(1,2), B(4,3), C(6,6) in order

The fourth corner

Diagonals are AC and BD, so their midpoints agree.

Takeaway. 'In order' names which vertices are opposite.

Which pairs are diagonals

'Taken in order' matters

In order, the diagonals join the 1st to the 3rd and the 2nd to the 4th.

Takeaway. Read the word 'order' in the question.

Three standard setups

Points with a condition

Takeaway. Writing the point's form first removes an unknown.

Choosing the tool

Which formula?

Length, perimeter, kind of triangle → DISTANCE.

Takeaway. Collinearity is distance, unsquared.

Justify the name

The sentence carries the marks

Write 'all four sides equal AND the diagonals equal, so it is a square'.

Takeaway. Show the six lengths in a list.

A note on coincidence

Squares are rare

Three random points are almost never collinear, and four are almost never a nice quadrilateral.

Takeaway. A question claiming they are is asking you to verify an exact fact.