MATH SOLVE

2 months ago

Q:
# *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

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