*URGENT* PLEASE 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
Accepted Solution
A:
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