Year 12 Coordinate Geometry: Planar shapes from a line

This subtopic starts with a known side and asks you to build the rest of a shape. The main idea is that a side vector contains both direction and length. Once that vector is known, you can create the other sides by translating it, rotating it, or using a perpendicular vector of equal or different length depending on the shape. These questions connect algebra and geometry very directly.

For a square or rectangle, rotate the side vector by 90 degrees to get a perpendicular side. For a parallelogram, translate one side vector to the opposite vertex.

Common shape rules

Square: adjacent sides are perpendicular and equal in length.
Rectangle: adjacent sides are perpendicular, opposite sides are parallel and equal.
Parallelogram: opposite sides are equal and parallel.
Rhombus: all sides equal, opposite sides parallel.

Worked example 1: building a square

Problem: Points

A (1, 2)

and

B (5, 3)

are adjacent vertices of a square

A B C D

. Find one possible set of coordinates for

C

and

D

The side vector is $vector A B = (4, 1)$ .
A perpendicular vector of the same length is $(- 1, 4)$ . Rotating $(4, 1)$ by 90 degrees gives this new direction.
Add this perpendicular vector to $B$ to get $C$ : $C = (5, 3) + (- 1, 4) = (4, 7)$ .
Add the same vector to $A$ to get $D$ : $D = (1, 2) + (- 1, 4) = (0, 6)$ .
Check. The new side vector $B C$ is $(- 1, 4)$ , which is perpendicular to $A B$ and has the same length, so the shape is a square.

Answer: one valid square is

A (1, 2)

B (5, 3)

C (4, 7)

D (0, 6)

Worked example 2: building a parallelogram

Problem: Suppose

A (2, 1)

B (7, 4)

and

D (4, 8)

are three vertices of parallelogram

A B C D

. Find

C

In a parallelogram, opposite sides are equal and parallel, so $vector B C = vector A D$ .
Find $vector A D = (4 - 2, 8 - 1) = (2, 7)$ .
Add this to $B$ : $C = (7, 4) + (2, 7) = (9, 11)$ .
You can also check with $vector D C$ . Since it should equal $vector A B = (5, 3)$ , adding that to $D (4, 8)$ also gives $(9, 11)$ .

Answer:

C = (9, 11)

How to think about perpendicular vectors

If a vector is $(a, b)$ , then a perpendicular vector can be $(- b, a)$ or $(b, - a)$ . Those two options correspond to the two possible orientations of the shape. That is why square and rectangle questions often have two valid answers unless the diagram or wording fixes the orientation.

Common mistakes

Using a perpendicular vector with the wrong length for a square.
Forgetting there are sometimes two possible shapes.
Mixing up which side should be translated in a parallelogram.
Not checking the final shape with slope, vector, or distance information.

Final check

Once you have candidate vertices, check the geometry. Are adjacent sides perpendicular when they should be? Are equal sides actually equal? Do opposite sides match as vectors? A short verification step is often the difference between a correct answer and a plausible-looking wrong one.

Practice links

Coordinates of a planar shape from a line