Question

Help me with 12th grade geometry! 30 points! Please explain as thoroughly as possible!

Answer 1

Answer:

b = 12

c = 8

d = 16

Step-by-step explanation:

The area of a parallelogram is the base times the height.  If we say AD is the base, then the height is the perpendicular distance from B to AD.

First, find the equation of AD.

m = (2 − (-2)) / (4 − (-2))

m = 2/3

y − (-2) = 2/3 (x − (-2))

y + 2 = 2/3 (x + 2)

y + 2 = 2/3 x + 4/3

3y + 6 = 2x + 4

-2x + 3y + 2 = 0

The perpendicular distance between the line -2x + 3y + 2 = 0 and the point (2, b) is:

h = |-2×2 + 3b + c| / √((-2)² + 3²)

h = (-4 + 3b + c) / √13

The length of the base, AD, is:

AD = √((4 − (-2))² + (2 − (-2))²)

AD = √(36 + 16)

AD = √52

AD = 2√13

Using the area of the parallelogram:

80 = (2√13) (-4 + 3b + c) / √13

80 = 2 (-4 + 3b + c)

40 = -4 + 3b + c

44 = 3b + c

Next, use the fact that opposite sides are parallel.

ABₓ = CDₓ

2 − (-2) = c − 4

4 = c − 4

c = 8

Plug in to find b:

44 = 3b + 8

36 = 3b

b = 12

Now find d:

ABᵧ = CDᵧ

b − (-2) = d − 2

12 + 2 = d − 2

d = 16