Find the area of a parallelogram bounded by the y-axis, the line x = 9, the line f(x) = 9 + 2x, and the line parallel to f(x) passing through (4, 16).

Question

Recall the formula for the area of a parallelogram. Consider sketching a graph of the given lines and point. Where does the given line intersect the y-axis (the line
x = 0)
and the line x = 9? Where does the parallel line that passes through the point (4, 16) intersect the y-axis and the line x = 9? Use these points to determine the height and base of the parallelogram.

Answer 1

These four lines when intersected will form a parallelogram. 

 

First, lets find the equation of the last line.  This line is parallel to f(x) and passes through the point (4, 16).  Lets call this line g(x).

 

g(x) = 2x + b

 

16 = 2(4) + b

 

16 = 8 + b

 

8 = b

 

 

g(x) = 2x + 8

 

 

Now that we have all the defined lines, draw them on the same coordinate system.  A parallelogram has four sides, so we should end up with four points of intersections that form the closed shape.  Then use the four points to find the find distance between the points.  Use the distance formula:

 

d = √[(x₁ - x₂)² + (y₁ - y₂)²]

 

 

Once you have the side lengths, you can get the area of the quadrilateral. 

 

Area = base × height