Question

Determine whether a figure with the given vertices is a rectangle using the distance formula.

Answer 1

`\begin{gathered} a)\text{ No},\text{ it is not a parallelogram} \\ b)\text{ length of AB = 6} \\ c)\text{ length of BC = 5} \\ d)\text{ length of CD = 6} \\ e)\text{ length of DA = 5} \\ f)\text{ yes, a rectangle} \\ g)\text{ length of diagonal AC = }\sqrt[]{61} \\ h)\text{ length of diagonal BD = }\sqrt[]{61} \end{gathered}`

Step-by-step explanation:

For the shape to be a parallelogram, its opposite sides will be equal. Also the diagonals will only bisect each other. They won't be equal

Let's check if the diagonals are equal. Using an illustration:

The diagonals from the image are AD and BC.

We will apply the distance formula to get the lengths of the diagonals:

`dis\tan ce\text{ = }\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2}`
`\begin{gathered} A\text{ = (-6, -5) and D = (-1, -5)} \\ \text{distance AD = }\sqrt[]{(-5-(-5))^2+(-1-(-6))^2} \\ \text{distance AD = }\sqrt[]{(-5+5)^2+(-1+6)^2}\text{ = }\sqrt[]{0^2+5^2} \\ \text{distance AD = }\sqrt[]{25}\text{ = 5} \\ \\ B\text{ = (-6, 1) and C = (-1, 1)} \\ \text{distance BC = }\sqrt[]{(1-1)^2+(-1-(-6))^2} \\ \text{distance BC = }\sqrt[]{(0)^2+(-1+6)^2}\text{ = }\sqrt[]{0+5^2} \\ \text{distance BC = }\sqrt[]{25}\text{ = 5} \end{gathered}`

a) The diagonals AD and BC have the same length.

Since the diagonals of a parallelogram is not the same. The quadrilateral ABCD is not a parallelogram

Ist blank: No

b) To get length of AB, we will apply the distance formula

`\begin{gathered} A\text{ = }(-6,\text{ -5) and B = (-6, 1)} \\ \text{distance AB = }\sqrt[]{(1-(-5))^2+(-6-(-6))^2} \\ \text{distance AB = }\sqrt[]{(1+5)^2+(-6+6)^2}\text{ = }\sqrt[]{6^2+0^2} \\ \text{distance AB = }\sqrt[]{36}\text{ = 6} \\ \text{length of side AB = 6} \end{gathered}`

length of side BC = distance BC

length of side BC = 5

c) Distance CD:

`\begin{gathered} C\text{ = (-1, 1) and D = (-1, -5)} \\ \text{distance CD = }\sqrt[]{(-5-1)^2+(-1-(-1))^2} \\ \text{distance CD = }\sqrt[]{(-6)^2+(-1+1)^2}\text{ = }\sqrt[]{36+0^2} \\ \text{distance CD = }\sqrt[]{36}\text{ = 6} \\ \text{distance CD = length of side CD} \\ \\ \text{ length of side CD = }6 \end{gathered}`

d) Distance DA = distance AD = 5

Distance DA = length of side DA

length of side DA = 5

f) For the quadrilateral to be a rectangle, its opposite sides will be equal. The diagonals will also be equal

Since the diagonals of the shape are equal.

Yes, the quadrilateral is a rectangle

length of diagonal AC:

`\begin{gathered} A\text{ = (-6, -5) and C = (-1, 1)} \\ \text{distance AC = }\sqrt[]{(1-(-5))^2+(-1-(-6))^2} \\ \text{distance AC = }\sqrt[]{(1+5)^2+(-1+6)^2}\text{ = }\sqrt[]{(6)^2+(5)^2} \\ \text{distance AC = }\sqrt[]{36\text{ + 25}}\text{ = }\sqrt[]{61} \\ dis\tan ce\text{ AC = length of AC} \\ \\ \text{length of diagonal AC = }\sqrt[]{61} \end{gathered}`

length of diagonal BD:

`\begin{gathered} B\text{ = }(-6,\text{ 1) and D = (-1, -5)} \\ \text{Distance BD = }\sqrt[]{(-5-1)^2+(-1-(-6))^2} \\ \text{Distance BD = }\sqrt[]{(-6)^2+(-1+6)^2}\text{ = }\sqrt[]{36\text{ + 25}}\text{ } \\ \text{Distance BD = }\sqrt[]{61} \\ \text{Distance BD = length of BD} \\ \\ \text{length of diagonal BD = }\sqrt[]{61} \end{gathered}`