Classifying Triangles & Quadrilaterals by Coordinates

When a figure's vertices are given as coordinates, you can classify it without a ruler or protractor — algebra does the measuring. Two formulas handle everything: the distance formula tells you how long each side is, and the slope formula tells you how each side is tilted.

The strategy is always the same. Decide what the classification requires — congruent sides, parallel sides, or a right angle — then compute exactly the distances or slopes that test it, and compare.

The two tools

Distance formula: the length of the segment from $(x_1, y_1)$ to $(x_2, y_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . Use it whenever the classification is about lengths — congruent sides, isosceles, equilateral, rhombus.

Slope formula: the slope of that same segment is $\displaystyle \frac{y_2 - y_1}{x_2 - x_1}$ . Use it whenever the classification is about direction — equal slopes mean parallel sides, and slopes that are opposite reciprocals (like $3$ and $\displaystyle -\frac{1}{3}$ ) mean perpendicular sides, which is how you find a right angle.

Leave lengths as radicals. Comparing $\sqrt{29}$ with $\sqrt{29}$ is exact; comparing rounded decimals can hide whether two sides are truly congruent.

Classifying triangles

Compute all three side lengths. Three congruent sides make the triangle equilateral, exactly two make it isosceles, and no congruent pair makes it scalene.

To test for a right triangle, compute the slopes of the sides meeting at the suspected right angle and check for opposite reciprocals. A horizontal side (slope $0$ ) and a vertical side (undefined slope) are also perpendicular. Classifications can stack — a triangle can be right and isosceles at the same time.

Classifying quadrilaterals

Walk the vertices in order around the figure and compute the slope of each side. Both pairs of opposite sides parallel makes a parallelogram. Add four right angles (adjacent slopes are opposite reciprocals) and it is a rectangle; add four congruent sides instead and it is a rhombus; both together make a square.

One pair of parallel sides — and only one — makes a trapezoid. The order you list the vertices matters: connect them as they sit around the figure, or you will compute a diagonal where a side should be.

Worked examples

Example 1: classify a triangle by side lengths

Classify the triangle with vertices $A(0, 0)$ , $B(6, 0)$ , and $C(3, 4)$ .

Find

AB

— a horizontal segment

AB = 6

Find

AC

with the distance formula

AC = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{25} = 5

Find

BC

with the distance formula

BC = \sqrt{(3 - 6)^2 + (4 - 0)^2} = \sqrt{25} = 5

Exactly two sides are congruent

\text{isosceles}

Answer: Isosceles

Example 2: test for a right angle with slopes

Show that $\triangle PQR$ with $P(1, 1)$ , $Q(4, 3)$ , and $R(2, 6)$ has a right angle at $Q$ .

Slope of

\overline{PQ}

using

\displaystyle \frac{y_2 - y_1}{x_2 - x_1}

\displaystyle \frac{3 - 1}{4 - 1} = \frac{2}{3}

Slope of

\overline{QR}

using

\displaystyle \frac{y_2 - y_1}{x_2 - x_1}

\displaystyle \frac{6 - 3}{2 - 4} = -\frac{3}{2}

The slopes are opposite reciprocals

\displaystyle \frac{2}{3} \cdot \left(-\frac{3}{2}\right) = -1

Perpendicular sides meet at a right angle at

Q

\text{right triangle}

Answer: $\overline{PQ} \perp \overline{QR}$ , so the triangle is right.

Example 3: identify a parallelogram

Classify the quadrilateral with vertices $(0, 0)$ , $(5, 1)$ , $(7, 5)$ , and $(2, 4)$ , taken in order.

Slope of the side from

(0, 0)

(5, 1)

\displaystyle \frac{1 - 0}{5 - 0} = \frac{1}{5}

Slope of the opposite side from

(2, 4)

(7, 5)

\displaystyle \frac{5 - 4}{7 - 2} = \frac{1}{5}

Slope of the side from

(5, 1)

(7, 5)

\displaystyle \frac{5 - 1}{7 - 5} = 2

Slope of the opposite side from

(0, 0)

(2, 4)

\displaystyle \frac{4 - 0}{2 - 0} = 2

Both pairs of opposite sides are parallel

\text{parallelogram}

Answer: Parallelogram

Try one yourself

Common questions

Should I simplify square roots or use decimals?

Keep exact radicals like $\sqrt{29}$ while comparing sides. Two sides are congruent only when the radicals match exactly — rounded decimals can make unequal sides look equal or equal sides look different.

When do I use distance and when do I use slope?

Distance answers length questions: congruent sides, isosceles, equilateral, rhombus. Slope answers direction questions: parallel sides, perpendicular sides, right angles. Many figures need both — a square, for example, needs congruent sides and right angles.

Can a triangle have two classifications at once?

Yes. Side-based names (scalene, isosceles, equilateral) and angle-based names (acute, right, obtuse) are independent, so a triangle can be a right isosceles triangle. Give both when a problem asks for the best description.

Does the order of the vertices matter for a quadrilateral?

Yes. List the vertices as you meet them walking around the figure. If you skip around, the segment you compute may be a diagonal instead of a side, and every slope comparison after that is wrong.

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Classifying Triangles & Quadrilaterals by Coordinates

The two tools

Classifying triangles

Classifying quadrilaterals

Worked examples

Example 1: classify a triangle by side lengths

Example 2: test for a right angle with slopes

Example 3: identify a parallelogram

Try one yourself

Common questions

Keep learning

Coordinate Proofs

The Distance & Midpoint Formulas on the Coordinate Plane

Calculating Slope from Two Points

Isosceles & Equilateral Triangle Theorems