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Answer:
- rectangle; parallelogram with 90° angle
- 150 feet
- $40 flat rate
Step-by-step explanation:
1. The slope of AB is ...
mAB = (yB-yA)/(xB-xA) = (4-0)/(6-3) = 4/3
The slope of AD is ...
mAD = (yD-yA)/(xD-xA) = (6-0)/(-5-3) = 6/-8 = -3/4
The product of these slopes is (4/3)(-3/4) = -1, so the segments AB and AD are perpendicular.
The midpoints of AC and BD will be the same if the sum of their coordinates is the same:
A+C = (3, 0) +(-2, 10) = (1, 10)
B+D = (6, 4) +(-5, 6) = (1, 10)
The midpoints of the diagonals of ABCD are the same, so the figure is a parallelogram. One corner angle is 90°, so the parallelogram is a rectangle. The quadrilateral ABCD is a rectangle.
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2. Using the coordinate differences found in part 1, we know the length AB is the hypotenuse of a 3-4-5 triangle, so is 5 grid squares. Similarly, the length of AD is twice that length, so is 10 grid squares. The perimeter in grid squares is ...
P = 2(L+W) = 2(10+5) = 30 . . . . grid squares
Each grid square represents 5 ft, so the perimeter is ...
(30 squares)(5 ft/square) = 150 ft . . . perimeter
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3. The rectangle is 5 × 10 grid squares, or 25 × 50 ft. Then its area is ...
A = LW = (50 ft)(25 ft) = 1250 ft²
At $0.03/ft², the payment for mowing would be ...
($0.03/ft²)(1250 ft²) = $37.50
The $40 flat rate would make you the most money.
