8.6.1 Finding and Reasoning Unknown Factors

Enter the missing area.

Compare your answer:

To determine the area of the blue rectangle, we need to figure out it’s width and length.

The top shape is a square measuring 8 in. x 8 in. If the measurement of the longer left side is 10 inches, then the width of the blue rectangle can be an equation.

$\begin{array}{ccc}\hfill 8+W& \hfill =\hfill & 10\hfill \\ \hfill W& \hfill =\hfill & 2\hfill \end{array}$

The width of the rectangle is 2 inches.

Because the bottom of the square is 8 inches, the length of the blue rectangle can be written as an equation. But we have an unknown $z$.

$L=8+z$

If we look at the measurements of the smaller square, we see that we can use an equation to calculate $z$.

$\begin{array}{ccc}\hfill z+3& \hfill =\hfill & 5\hfill \\ \hfill z& \hfill =\hfill & 2\hfill \end{array}$

Now, we can substitute the 2 for $z$ in the length equation.

$\begin{array}{ccc}\hfill L& \hfill =\hfill & 8+2\hfill \\ \hfill L& \hfill =\hfill & 10\hfill \end{array}$

The area of the blue rectangle is $8x10$to equal 80 square inches.

Enter the missing length.

Compare your answer:

You have to start with what you know and can figure out the other sides.

Sides of orange rectangle

$\begin{array}{ccc}\hfill L·W& \hfill =\hfill & Area\hfill \\ \hfill 12W& \hfill =\hfill & 60\hfill \\ \hfill W& \hfill =\hfill & 5\hfill \end{array}$

Sides of green rectangle

$\begin{array}{ccc}\hfill L·W& \hfill =\hfill & Area\hfill \\ \hfill (5+3)·W& \hfill =\hfill & 48\hfill \\ \hfill 8w& \hfill =\hfill & 48\hfill \\ \hfill w& \hfill =\hfill & 6\hfill \end{array}$

Sides of combined rectangles

The measurement of the long side of combined orange and green rectangle will be equal. This helps figure out one side of the blue rectangle.

$\begin{array}{ccc}\hfill 12+6& \hfill =\hfill & 3+w+11\hfill \\ \hfill 18& \hfill =\hfill & w+14\hfill \\ \hfill w& \hfill =\hfill & 4\hfill \end{array}$

Sides of blue rectangle

$\begin{array}{ccc}\hfill L·W& \hfill =\hfill & Area\hfill \\ \hfill L·4& \hfill =\hfill & 36\hfill \\ \hfill L& \hfill =\hfill & 9\hfill \end{array}$