Distance and the Pythagorean Theorem Practice

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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.

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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 derived from the Pythagorean theorem.

The coordinates of two opposite corners of the rectangle are \((-6, 8)\) (top left) and \((4, -3)\) (bottom right).

1. Calculate the difference in the x-coordinates: \[ \text{Difference in x} = x_2 - x_1 = 4 - (-6) = 4 + 6 = 10 \]

2. Calculate the difference in the y-coordinates: \[ \text{Difference in y} = y_2 - y_1 = -3 - 8 = -3 - 8 = -11 \quad \text{(take the absolute value)} \quad = 11 \]

3. Now use the Pythagorean theorem to find the length of the diagonal \(d\): \[ d = \sqrt{(\text{Difference in x})^2 + (\text{Difference in y})^2} \] \[ d = \sqrt{(10)^2 + (11)^2} = \sqrt{100 + 121} = \sqrt{221} \]

4. Calculate \( \sqrt{221} \): Using a calculator, we find: \[ \sqrt{221} \approx 14.83 \]

Thus, the length of the diagonal of the rectangle, rounded to the nearest hundredth, is approximately \( \boxed{14.83} \) units.