Given the rectangle ABCD on the coordinate plane. The following of the vertices of the rectangle: A = (0,0) B = (4,4) C = (7,-1) D = (3,3)
Fill in the blanks.
The length of side AB in simplest radical form is blank
The length of side BC in simplest radical form is blank
The area of the rectangle ABCD in simplest radical form is blank
Answer choices
√32, 4√2, 2√4, √18, 3√2, 9√2, 24, 14√2
To fill in the blanks, we will calculate the lengths of the sides AB and BC using the distance formula, which for two points (x1, y1) and (x2, y2) is √[(x2 - x1)^2 + (y2 - y1)^2].
Length of side AB:
A = (0,0), B = (4,4)
AB = √[(4 - 0)^2 + (4 - 0)^2] = √[16 + 16] = √32
Length of side BC:
B = (4,4), C = (7,-1)
BC = √[(7 - 4)^2 + (-1 - 4)^2] = √[3^2 + (-5)^2] = √[9 + 25] = √34 (which is not an answer choice, indicating there might be an error in the question or the provided points do not form a rectangle as defined)
To calculate the area of the rectangle, we multiply the lengths of two adjacent sides. However, as noted above, the points given do not form a rectangle where the sides meet at right angles, which means the calculation of the area using these points is not appropriate for a rectangle.
If we were to incorrectly proceed with the assumption that the shape is a rectangle, we would traditionally take the product of AB and BC (with the correct value for BC). Unfortunately, since BC is calculated as √34 based on the given points, which isn't in the answer choices, we can't compute the required area for this "rectangle."
From the answer choices provided, none match the lengths calculated using the distance formula for the coordinates given. Please verify the coordinates for rectangle ABCD; there might be a typo or mistake in the provided vertices. If ABCD is a true rectangle, then sides AB and BC should be perpendicular, and their lengths should correspond to the answer choices given.