To find the length of the diagonal of the rectangle, we can use the distance formula. The distance formula is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, we can treat the diagonal of the rectangle as the hypotenuse of a right triangle, with the sides of the rectangle as the other two sides.
The length of the rectangle's side is the difference between the x-coordinates of two adjacent points, and the width of the rectangle is the difference between the y-coordinates of two adjacent points.
Let's calculate the length of the diagonal:
1. Calculate the length of the side of the rectangle:
- The x-coordinates of the two adjacent points are -6 and 4, so the difference is 4 - (-6) = 10.
- The y-coordinates of the two adjacent points are 8 and 8, so the difference is 8 - 8 = 0.
- The length of the side of the rectangle is 10 units.
2. Calculate the width of the rectangle:
- The x-coordinates of the two adjacent points are 4 and 4, so the difference is 4 - 4 = 0.
- The y-coordinates of the two adjacent points are 8 and -3, so the difference is -3 - 8 = -11 (we take the difference in magnitude, ignoring the negative sign).
- The width of the rectangle is 11 units.
3. Calculate the diagonal of the rectangle using the Pythagorean theorem:
- The length of the side is 10 units.
- The width of the rectangle is 11 units.
To find the length of the diagonal, we can apply the Pythagorean theorem:
diagonal^2 = side^2 + width^2
diagonal^2 = 10^2 + 11^2
diagonal^2 = 100 + 121
diagonal^2 = 221
Taking the square root of both sides gives us the length of the diagonal:
diagonal = sqrt(221) ≈ 14.87 units
Therefore, the length of the diagonal of the rectangle is approximately 14.87 units.
Distance and the Pythagorean Theorem Practice
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Question
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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.
(1 point)
units
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