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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)
units

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 distance formula:

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

We can choose two opposite corners of the rectangle to calculate the diagonal. Let's use the points \((-6, 8)\) and \((4, -3)\).

1. Identify the coordinates of the points:

   • Point 1: \((-6, 8)\) which is \((x_1, y_1)\)
   • Point 2: \((4, -3)\) which is \((x_2, y_2)\)

2. Substitute these values into the distance formula:

\[ d = \sqrt{(4 - (-6))^2 + (-3 - 8)^2} \]

3. Simplify the equations:

   • \(x_2 - x_1 = 4 - (-6) = 4 + 6 = 10\)
   • \(y_2 - y_1 = -3 - 8 = -11\)

4. Now, substitute these into the formula:

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

5. Calculate \(\sqrt{221}\):

\[ \sqrt{221} \approx 14.8661 \]

6. Round to the nearest hundredth:

\[ \sqrt{221} \approx 14.87 \]

Thus, the length of the diagonal of the rectangle is approximately 14.87 units.