Polygons can be classified based on their side lengths and angle measures.

For triangles, we have the following classifications:

Note that it is possible for a triangle to be a scalene right triangle or an isosceles right triangle.

Triangles that are not right triangles can be further classified as acute triangles (if all angle measures are smaller than 90 \degree) or obtuse triangles (if one angle measure is larger than 90 \degree).

For quadrilaterals, we have the following classifications:

We want to classify polygons as precisely as possible:

If we know the coordinates of the vertices of a polygon, we can find the most precise classification using the lengths and slopes of line segments joining pairs of vertices.

- Find the slope of each side to determine if any sides are parallel or perpendicular.
- Find the length of each side to determine if any sides are congruent.

Consider the given quadrilateral.

a

Determine the slopes of \overline{AB}, \overline{BC}, \overline{CD}, and \overline{AD}.

Worked Solution

b

Calculate the lengths of the sides.

Worked Solution

c

Classify the quadrilateral as precisely as possible.

Worked Solution

Classify the triangle with vertices at A \left(-6,-2\right), B \left(-4,4\right), and C \left(14,-2\right) as precisely as possible.

Worked Solution

Show that the quadrilateral with vertices at A\left(2,1\right),\,B\left(1,4\right),\,C\left(6,5\right),\, D\left(7,2\right) is not a rectangle.

Worked Solution

Let ABCD be a rhombus with vertices at A\left(-2,-1\right),\, B\left(x,y\right),\, C\left(-2,-7\right) and D\left(-6,-4\right).

a

Find the coordinates of vertex B.

Worked Solution

b

Verify ABCD is a rhombus.

Worked Solution

Idea summary

If we know the coordinates of the vertices of a polygon, we can find the most precise classification using the lengths and slopes of line segments joining pairs of vertices.

- Find the slope of each side to determine if any sides are parallel or perpendicular.
- Find the length of each side to determine if any sides are congruent.