Coordinate Geometry

[Pawan Ramnani]

What is the area of the area of the triangle formed by the intersection of y = 2x-2, y = -1/2x + 8 and y=0 ?

a)45
b)67
c)89
d)31
e)37

[MartyMurray]

The most straightforward way to find the area of a triangle is by multiplying base * height and dividing by 2.

So if there is a not too challenging way to get a base and height of this triangle, we are set.

Here are the three lines that intersect to form the triangle.

y = 2x - 2

y = -1/2x + 8

y = 0

y = 0 is the x axis and is a horizontal line, so the side of the triangle that lies on this line seems to be a great choice for the base. Now we need the length of the base.

y = 2x - 2 intersects the x axis when y = 0. y = 0 when x = 1.

y = -1/2x + 8 intersects the x axis when x = 16.

So the base of the triangle goes from point (1,0) to point (16,0) and is 16 - 1 = 15 long.

To get the height. first find the other vertex of the triangle.

Set the two equations, y = 2x - 2 and y = -1/2x + 8 equal to each other.

2x - 2 = -1/2x + 8

2 1/2x = 10

x = 4

y = 2(4) - 2 = 6

Since the base is at y = 0 and the y coordinate of the other vertex is 6, the height is 6.

bh/2 = (15*6)/2 = 45

The correct answer is A.

[Matt@VeritasPrep]

The three vertices of the triangle will be where the three lines intersect, so let's find those coordinates.

First intersection: y = 0 and y = 2x - 2

Since y = 0, x = 1, and this coordinate is (1, 0).

Second intersection: y = 0 and y = -.5x + 8

Since y = 0, x = 16, and this coordinate is (16, 0).

This gives us a base of length 15, since (1,0) to (16,0) is a flat line with length 15 (16 - 1).

From there, we'll find the last coordinate:

Third intersection: y = 2x - 2, y = -.5x + 8

Solving the two equations, we get x = 4, y = 6.

The height of the triangle is the length from the base (y = 0) to this point (y = 6), so the height is 6.

We've got base 15 and height 6, so our area is 6*15/2, or 45.