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 find the lengths of AB and BC, we can use the distance formula between two points on the coordinate plane:
Distance AB = √[(x2 - x1)^2 + (y2 - y1)^2]
Distance BC = √[(x2 - x1)^2 + (y2 - y1)^2]
Given A = (0,0) and B = (4,4), we can calculate distance AB:
AB = √[(4 - 0)^2 + (4 - 0)^2]
= √(16 + 16)
= √32
Given B = (4,4) and C = (7,-1), we can calculate distance BC:
BC = √[(7 - 4)^2 + (-1 - 4)^2]
= √(3^2 + (-5)^2)
= √(9 + 25)
= √34
However, the provided answer choices don't include √34, so we would need to double-check the coordinates. If the vertices are meant to be the corners of a rectangle, point D should be 3 units in the x direction and 3 units in the negative y direction from point A to be the fourth vertex. The given point D at (3,3) is not consistent with the definition of a rectangle where opposite sides must be equal in length and parallel.
Given C = (7,-1) and assuming D = (3,-1) to correct the coordinate for a rectangle, we can calculate distance CD (which should be the same as AB if ABCD is a rectangle):
CD = √[(7 - 3)^2 + (-1 - (-1))^2]
= √(4^2 + 0^2)
= √16
= 4
So we see that the correct distance BC (assuming the rectangle) is 4 units.
With these distances, we now have:
The length of side AB in simplest radical form is √32.
The length of side BC (corrected) in simplest radical form is 4 (or we can express this as 2√4, given the answer choices).
To find the area of rectangle ABCD, we multiply the lengths of AB and BC:
Area of ABCD = AB × BC
= √32 × 4
= √(32) × √(4)
= 4√8
= 4(2√2)
= 8√2
However, there's something off with the answers provided and the scenario described. The points given do not form a rectangle since the answer should fit within the provided choices and none of the sides are parallel to the coordinate axes. To reconcile the observed lengths with the answer choices, it seems there has been a mistake in the coordinates provided or with the answer options. If we assume a typo and correct the coordinates of point D to (3, -1), we get:
Length of side AB is √32, which is correct.
Length of side BC (which was 4, but possibly should match an answer choice) could be seen as 2√4.
Area of the rectangle ABCD, using the corrected side lengths, would be √32 × 2√4 (or 4×4 based on our corrected BC).
Without clarification or correction of either points or answer choices, we can't provide one of the given choices as the definitive answer. If the figure was actually a rectangle, and the coordinate of point D was indeed a mistake, this analysis may help clear out the misconceptions.