Question

Given the set of vertices, determine whether parallelogram ABCD is a rhombus, rectangle or square. List all that apply. A(7,-4), B(-1,-4), C(-1,-12), D(7, -12)

a. rhombus c. square, rectangle, rhombus
b. square d. rectangle

Answer 1

Given:

Vertices of a parallelogram ABCD are A(7,-4), B(-1,-4), C(-1,-12), D(7, -12).

To find:

Whether the parallelogram ABCD is a rhombus, rectangle or square.

Solution:

Distance formula:

`D=√((x_2-x_1)^2+(y_2-y_1)^2)`

Using distance formula, we get

`AB=√((-4-(-4))^2+(-1-7)^2)`

`AB=√((-4+4)^2+(-8)^2)`

`AB=√(0+64)`

`AB=8`

Similarly,

`BC=√((-1-(-1))^2+(12-(-4))^2)=8`

`CD=√((7-(-1))^2+(-12-(-12))^2)=8`

`AD=√((7-7)^2+(-12-(-4))^2)=8`

All sides of parallelogram are equal.

`AC=√((-1-7)^2+(-12-(-4))^2)=8√(2)`

`BD=√((7-(-1))^2+(-12-(-4))^2)=8√(2)`

Both diagonals are equal.

Since, all sides are equal and both diagonals are equal, therefore, the parallelogram ABCD is a square.

We know that, a square is special case of rectangles and rhombus.

So, parallelogram ABCD is a rhombus, rectangle or square. Therefore, the correct option is c.