Expert: Sherman D. - 9/16/2006

Question
the line 3x - 4y=8 meets the y axis at A. The point C has co-ordinates (-2,9). the line through C perpendicular to 3x - 4y+8 meets it at B. calculate the area of the triangle.

Answer
3x - 4y = 8
-4y = -3x + 8
y = (3/4)x - 2

(-2,9), m = (-4/3)
9 = (-4/3)(-2) + b
9 = (8/3) + b
27 = 8 + 3b
3b = 19
b = (19/3)
y = (-4/3)x + (19/3)

(3/4)x - 2 = (-4/3)x + (19/3)
9x - 24 = -16x + 76
25x = 100
x = 4

y = (-4/3)(4) + (19/3)
y = (-16/3) + (19/3)
y = (-16 + 19)/3
y = (3/3)
y = 1

So the 2 lines meet at (4,1)

A = (0,-2)
B = (4,1)
C = (-2,9)

Distance formula
D = sqrt((x2 - x1)^2 + (y2 - y1)^2)

(0,-2) and (4,1)
D = sqrt((4 - 0)^2 + (1 - (-2))^2)
D = sqrt((4)^2 + (1 + 2)^2)
D = sqrt(16 + 3^2)
D = sqrt(16 + 9)
D = sqrt(25)
D = 5

(4,1) and (-2,9)
D = sqrt((-2 - 4)^2 + (9 - 1)^2)
D = sqrt((-2 + (-4))^2 + 8^2)
D = sqrt((-6)^2 + 64)
D = sqrt(36 + 64)
D = sqrt(100)
D = 10

(0,-2) and (-2,9)
D = sqrt((-2 - 0)^2 + (9 - (-2))^2)
D = sqrt((-2)^2 + (9 + 2)^2)
D = sqrt(4 + 11^2)
D = sqrt(4 + 121)
D = sqrt(125)
D = sqrt(25 * 5)
D = 5sqrt(5)

if you use a^2 + b^2 = c^2, you will find out that this is a right triangle

A = (ab)/2
A = (10 * 5)/2
A = 50/2
A = 25

ANS : the area is 25 square units