jmc

Question

A line with slope of -2 intersects the positive x-axis at A and the positive y-axis at B. A second line intersects the x-axis at C(8,0) and the y-axis at D. The lines intersect at E(4,4). What is the area of the shaded quadrilateral OBEC? FIGURE

Accepted Answer

First, we can create a square with the points

O

and

E

as opposite corners. Label the other two points as

X

and

Y

with

X

on

O C

and

Y

on

O B

. We get that

X

is

(4, 0)

and

Y

is

(0, 4)

.

We can find the area of the figure by finding the area of the square and the two triangles created.

The area of the square is

4 \cdot 4 = 16.

The two triangles are right triangles. The first one,

X C E

, has legs

X C

and

X E

of length

4

, so the area is

\frac{4 \cdot 4}{2} = 8

. To find the area of the other triangle, we must find the coordinates of

B (0, y)

. The slope of

B E

is of slope

- 2

. Therefore,

\frac{y - 4}{0 - 4} = - 2

.

Solving for

y

, we get

y = 12.

Then, the leg of the second triangle

B Y

is

12 - 4 = 8

. The area of the triangle

Y E B

is thus

\frac{8 \cdot 4}{2} = 16.

Adding the areas of the three areas together,

16 + 16 + 8 = 40 .