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
