Question

Can someone help me match the set of vertices with the type of quadrilateral they form?
A(2, 0), B(3, 2), C(6, 3), D(5, 1)

a parallelogram with
nonperpendicular adjacent sides

A(3, 3), B(3, 6), C(7, 6), D(7, 3)

a rectangle with noncongruent
adjacent sides

A(-5, -3), B(-4, -1), C(-1, -1), D(0, -3)

a square

A(2, -2), B(3, 0), C(4, -2), D(3, -4)
a rhombus with nonperpendicular
adjacent sides

A(3, 3), B(2, 5), C(4, 6), D(5, 4)

Answer 1

Try taking one of the sets of points and graphing them... See what you get and see which of the explanations of figures matches the set of points.

Answer 2

As an example consider the first set of points:

A(2, 0), B(3, 2), C(6, 3), D(5, 1). When graphed it looks like this:

Answer 3

In this case, thanks to @mathmate, it is a Rhombus.

Answer 4

It is not a rhombus because not all sides are congruent.

Answer 5

Right, it's a parallelogram. I was looking at it in small scale and thought "it looks like a rhombus".

Answer 6

If you prefer to work with the coordinates, you can do so using three simple tests,

1. to test if it is a parallelgram: check if \(x_a+x_c=x_b+x_d\)

2. to test if adjacent sides are equal (for squares and rhombuses)
check if \((x_b−x_a)^2+(y_b−y_a)^2=(x_c−x_b)^2+(y_c−y_b)^2\)
Note: you don't even need to square the numbers if the numbers are like
\((−3)^2+4^2\) and \(4^2+3^2\) because we know that after squaring, the sums will be identical.

3. to test if adjacent sides are perpendicular (for squares, rectangles)
check if \(\frac{yb−ya}{xb−xa}\times \frac{yc−yb}{xc−xb}=−1\)

Once you have the information, you can determine which type of quadrilateral it is, among parallelogram, rhombus, square and rectangles.
For examples and proofs, see:


http://mathm8.altervista.org/geometry/index.html