Question

Alana is writing a coordinate proof to show that the diagonals of a rectangle are congruent. She starts by assigning coordinates to a rectangle as shown. Then she uses the coordinates to write and simplify expressions for the lengths of the diagonals.

What is the length of one of the diagonals of the rectangle?

Answer 1

Explanation:

The diagonals of a rectangle are formed by connecting opposite corners. In this case, the diagonals would be AC and BD.

Using the distance formula, we can calculate the length of AC:

Distance AC = sqrt((x2 - x1)^2 + (y2 - y1)^2)

AC = sqrt((0 - 0)^2 + (0 - 5)^2)

AC = sqrt(0 + 25)

AC = sqrt(25)

AC = 5

So, the length of diagonal AC is 5.

We can also calculate the length of BD using the same formula:

BD = sqrt((7 - 0)^2 + (0 - 5)^2)

BD = sqrt(49 + 25)

BD = sqrt(74)

So, the length of diagonal BD is sqrt(74).

Therefore, the length of one of the diagonals of the rectangle is 5 units.

Answer 2

`\sf √(74) units`

Explanation:

In order to find the diagonal of a rectangle using coordinates, we can use the distance formula

The distance formula is used to calculate the distance between two points in a coordinate plane. The formula is as follows:

`\sf distance = √((x_2 - x_1)^2 + (y_2 - y_1)^2)`

where:

(x1, y1) and (x2, y2) are the coordinates of the two points

In this case, we are calculating the distance between the diagonals of the rectangle.

The coordinate of diagonal of rectangle is either (0, 5) and (7,0) or (7,5) and (0,0).

We need to choose one of them, so,

I will choose, (0,5) and (7,0)

Substituting these values into the distance formula, we get:

`\sf distance = √((7 - 0)^2 + (5 - 0)^2)`

`\sf distance = √(7^2 + 5^2)`

`\sf distance = √(49+25)`

`\sf distance = √(74)`

Therefore, the length of one of the diagonals of the rectangle is
`\sf √(74) units`