*urgent* answer a) calculate the perimeter of the excavation. b) determine the equation of the diagonal adc) use algebraic methods to show that ad is parallel to bc

A) To calculate the perimeter we must first calculate the lengths of each of the sides:
AB = 2 - (-4) = 6
AE = 4 - (-2) = 6
ED = 5 - (-4) = 9
CD = 2 - (-2) = 4

Now BC is a little trickier and we need to use the formula for the distance between two points: d = sq.root of((x2 - x1)^2 + (y2 - y1)^2)
Given that B has coordinates (2, 4) and C has coordinates (5, 2):
d = sq.root of((5 - 2)^2 + (2 - 4)^2)
= sq.root of (3^2 + (-2)^2)
= sq.root of(9 + 4)
= sq.root of(13)
= 3.606 (to three decimal places)

Perimeter = 6 + 6 + 9 + 4 + 3.606
= 28.606 m
(Or if we kept it as an exact value it would be 25 + sq.root of(13) m)

b) First we need to calculate the gradient of the line AD, using m = (y2 - y1)/(x2 - x1).
Given that A has coordinates (-4, 4) and D has coordinates (5, -2):
m = (-2 - 4)/(5 - (-4)
= -6/9
= -2/3

Now we can substitute this into the point-slope equation form y - y1 = m(x - x1) with another point, let's take A(-4, 4):
y - 4 = -2/3(x - (-4)
y = (-2/3)x - 8/3 + 4
y = (-2/3)x + 4/3
(This can also be rewritten as 3y + 2x = 4)

c) If AD is parallel to BC, then their gradients will be equal. First we need to find the gradient of BC using the same method that we used above, where m = (y2 - y1)/(x2 - x1)
Given that B has coordinates (2, 4) and C has coordinates (5, 2):
m = (2 - 4)/(5 - 2)
= -2/3
The gradient of AD = -2/3
Gradient of AD = gradient of BC = -2/3, therefor the two lines are parallel