Question

A graph shows four quadrants with the x-axis and y-axis both ranging from negative 10 to positive 10 in increments of 1. Unlabeled points are plotted at left parenthesis negative 6 comma 8 right parenthesis, left parenthesis 4 comma 8 right parenthesis, left parenthesis 4 comma negative 3 right parenthesis, and left parenthesis negative 6 comma negative 3 right parenthesis. A straight line joins the four points forming a rectangle.

What is the length of the diagonal of the rectangle? Round your answer to the nearest hundredth, if necessary.

(1 point)

Answer 1

To find the length of the diagonal of the rectangle formed by the points \((-6, 8)\), \((4, 8)\), \((4, -3)\), and \((-6, -3)\), we can use the coordinates of two opposite corners of the rectangle.

Let's choose the points \((-6, 8)\) and \((4, -3)\) as the opposite corners. The length of the diagonal can be calculated using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates of the chosen points into the formula:

- \(x_1 = -6\)
- \(y_1 = 8\)
- \(x_2 = 4\)
- \(y_2 = -3\)

Now substitute:

\[
d = \sqrt{(4 - (-6))^2 + (-3 - 8)^2}
\]
\[
d = \sqrt{(4 + 6)^2 + (-3 - 8)^2}
\]
\[
d = \sqrt{(10)^2 + (-11)^2}
\]
\[
d = \sqrt{100 + 121}
\]
\[
d = \sqrt{221}
\]

Now, we compute \(\sqrt{221}\):

\[
\sqrt{221} \approx 14.866
\]

Rounding to the nearest hundredth, the length of the diagonal of the rectangle is:

\[
\boxed{14.87}
\]