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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.
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.
Answers
Answered by
GPT-4o mini
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.
First, we need to determine the coordinates of two opposite corners of the rectangle. We can select the points \( (-6, 8) \) and \( (4, -3) \).
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates \( (-6, 8) \) and \( (4, -3) \):
- \( x_1 = -6 \), \( y_1 = 8 \)
- \( x_2 = 4 \), \( y_2 = -3 \)
Now plug these values into the formula:
\[
d = \sqrt{(4 - (-6))^2 + (-3 - 8)^2}
\]
\[
d = \sqrt{(4 + 6)^2 + (-3 - 8)^2}
\]
\[
d = \sqrt{(10)^2 + (-11)^2}
\]
\[
d = \sqrt{100 + 121}
\]
\[
d = \sqrt{221}
\]
Next, we compute the value of \( \sqrt{221} \):
\[
\sqrt{221} \approx 14.866
\]
Rounding this to the nearest hundredth, we find:
\[
\sqrt{221} \approx 14.87
\]
Thus, the length of the diagonal of the rectangle is approximately \( \boxed{14.87} \).
First, we need to determine the coordinates of two opposite corners of the rectangle. We can select the points \( (-6, 8) \) and \( (4, -3) \).
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates \( (-6, 8) \) and \( (4, -3) \):
- \( x_1 = -6 \), \( y_1 = 8 \)
- \( x_2 = 4 \), \( y_2 = -3 \)
Now plug these values into the formula:
\[
d = \sqrt{(4 - (-6))^2 + (-3 - 8)^2}
\]
\[
d = \sqrt{(4 + 6)^2 + (-3 - 8)^2}
\]
\[
d = \sqrt{(10)^2 + (-11)^2}
\]
\[
d = \sqrt{100 + 121}
\]
\[
d = \sqrt{221}
\]
Next, we compute the value of \( \sqrt{221} \):
\[
\sqrt{221} \approx 14.866
\]
Rounding this to the nearest hundredth, we find:
\[
\sqrt{221} \approx 14.87
\]
Thus, the length of the diagonal of the rectangle is approximately \( \boxed{14.87} \).
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