Part A
The yard is a quadrilateral with the following corner points
I'll label those points A through D in the order presented above.
From A to B is 40 feet since 40-0 = 40. We can subtract the y coordinates because the x coordinates are the same.
In short: side AB is 40 feet long.
The length of side BC is not as simple. We'll need the distance formula.
[tex]B = (x_1,y_1) = (0,40) \text{ and } C = (x_2, y_2) = (40,50)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(0-40)^2 + (40-50)^2}\\\\d = \sqrt{(-40)^2 + (-10)^2}\\\\d = \sqrt{1600 + 100}\\\\d = \sqrt{1700}\\\\d = \sqrt{100*17}\\\\d = \sqrt{100}*\sqrt{17}\\\\d = 10\sqrt{17}\\\\d \approx 41.2311\\\\[/tex]
Segment BC is exactly [tex]10\sqrt{17}[/tex] feet long, which is about 41.2311 feet.
Use the distance formula again to find the distance from point C(40,50) to D(50,0)
[tex]C = (x_1,y_1) = (40,50) \text{ and } D = (x_2, y_2) = (50,0)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(40-50)^2 + (50-0)^2}\\\\d = \sqrt{(-10)^2 + (50)^2}\\\\d = \sqrt{100 + 2500}\\\\d = \sqrt{2600}\\\\d = \sqrt{100*26}\\\\d = \sqrt{100}*\sqrt{26}\\\\d = 10\sqrt{26}\\\\d \approx 50.9902\\\\[/tex]
Lastly, the side AD is exactly 50 feet because 50-0 = 50. We can subtract x coordinates because the y coordinates are the same. Use of the distance formula from A to D should show a result of 50 exactly.
We have these side lengths:
AB = 40
BC = 41.2311 (approximate)
CD = 50.9902 (approximate)
AD = 50
Add up those four sides and we'll get the perimeter of the quadrilateral.
AB+BC+CD+AD = 40+41.2311+50.9902+50 = 182.2213
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Part B
The garden is a rectangle that is 15 feet across horizontally (since subtracting x coordinates gives us 25-10 = 15) and 20 feet vertically (since subtracting y coordinates gives us 35-15 = 20).
This 15 by 20 rectangle has an area of 15*20 = 300 square feet.
The deck is a trapezoid with the parallel bases of 20 feet up top and 35 feet down below. Like before, we subtract x coordinates to find the horizontal distance (since the y coordinates are the same). The height of this trapezoid is 15 feet. Subtract the y coordinates to find the height.
The area of the trapezoid is
A = h*(b1+b2)/2
A = 15*(20+35)/2
A = 412.5
That decimal value is exact.
Add it onto the area of the rectangle
412.5+300 = 712.5
