Year 12 Coordinate Geometry: Distance and midpoint checks

Distance and midpoint formulas are often used as checking tools rather than starting tools. Once you have coordinates for vertices or image points, these formulas let you test whether the geometry matches the description. Do the diagonals bisect each other? Are two sides equal? Is a point exactly halfway between two endpoints? These questions appear repeatedly in coordinate proofs and verification problems.

Midpoint checks are especially useful for diagonals in parallelograms, while distance checks confirm equal sides, symmetry, and exact lengths.

Main formulas

Distance between $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ : $sqrt ({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})$
Midpoint of a segment: $((x_{1} + x_{2}) / 2, (y_{1} + y_{2}) / 2)$

When each formula is useful

Use midpoint when checking whether diagonals bisect each other.
Use distance when checking equal sides, radius lengths, or symmetry.
Use both together in problems about squares, rectangles, rhombuses, and parallelograms.

Worked example 1: proving a parallelogram

Problem: Show that the quadrilateral with vertices

A (1, 1)

B (5, 3)

C (7, 7)

and

D (3, 5)

is a parallelogram.

Find the midpoint of diagonal $A C$ : $((1 + 7) / 2, (1 + 7) / 2) = (4, 4)$ .
Find the midpoint of diagonal $B D$ : $((5 + 3) / 2, (3 + 5) / 2) = (4, 4)$ .
The diagonals have the same midpoint, so they bisect each other.
A quadrilateral whose diagonals bisect each other is a parallelogram.

Answer: the quadrilateral is a parallelogram because both diagonals share the midpoint

(4, 4)

Worked example 2: checking equal sides

Problem: Determine whether triangle

A B C

with

A (- 2, 1)

B (4, 5)

and

C (6, - 1)

is isosceles.

Find $A B$ : $sqrt ({(4 - (- 2))}^{2} + {(5 - 1)}^{2}) = sqrt (6^{2} + 4^{2}) = sqrt (52)$ .
Find $B C$ : $sqrt ({(6 - 4)}^{2} + {(- 1 - 5)}^{2}) = sqrt (2^{2} + {(- 6)}^{2}) = sqrt (40)$ .
Find $A C$ : $sqrt ({(6 - (- 2))}^{2} + {(- 1 - 1)}^{2}) = sqrt (8^{2} + {(- 2)}^{2}) = sqrt (68)$ .
No two side lengths are equal, so the triangle is not isosceles.

Answer: triangle

A B C

is not isosceles.

Why exact form matters

When using distance, it is usually better to compare squared lengths or exact square-root forms rather than decimals. For example, $sqrt (52)$ and $2 sqrt (13)$ are exactly equal, while decimal approximations may hide that relationship. In exam work, exact comparisons are clearer and safer.

Common mistakes

Dropping a negative sign when subtracting coordinates.
Comparing approximate decimals instead of exact values.
Using a midpoint argument when the problem really needs equal lengths.
Checking one feature of a shape and assuming the rest automatically follows.

Final check

After computing a midpoint or distance, always connect the result back to the geometry. Do not stop at "the midpoint is (4, 4)". State why that matters. Does it prove bisection, symmetry, equal radius, or some other property? The coordinates are only the middle step. The conclusion is the real mathematical goal.