Question

What fraction of the rectangle is shaded? Write your answer as a rational expression. Can you show the steps as well so I can understand it?

Answer 1

Well basically, the first thing to note is that anytime you are expressing a proper fraction, you are expressing part of a whole. In fraction form, it would look like this:

\[\frac{\text{Part}}{\text{Whole}}\]

In this case, we wish to express:

\[\frac{\text{Area of the Shaded Triangle}}{\text{Area of the Rectangle}}\]

In algebraic form.

Answer 2

If you recall, the Area of a rectangle is length times width or
\[A_{\text{rect}} = l \times w\]

The Area of a triangle is half its base times the height

\[A_{\triangle} = \frac{bh}{2}\]

Answer 3

In this case the length and width of the rectangle is:

l = x + 2
w = x

As it applies to rectangles, he length is ALWAYS the longest side, so x + 2 is the default length and x is the default width.

Answer 4

Yep

Answer 5

Notice that the base of the triangle corresponds to the length of the rectangle so

l = b = x + 2

Answer 6

The height of the triangle is 2 so

h = 2

Answer 7

Thus you have all the values needed to plug into both Area formulas and simplify. Let me know what you get.

Answer 8

Ok, so since the area of a triangle is bh/2 that means, x * 2/2, which is 2x/2 = x? Then the area of the rectangle is x (x + 2) = x^2 + 2x

Answer 9

It is best for you to express it in this manner just for absolute clarity and logic of the steps:

\[\frac{\text{Area of the Shaded Triangle}}{\text{Area of the Rectangle}} = \frac{\frac{bh}{2}}{lw} = \frac{\frac{2x}{2}}{x(x + 2)} = \frac{x}{x(x + 2)} = \frac{1}{x + 2}\]

Answer 10

Ah, makes sense, so you but the triangle as the numerator, but why? That is where I am confused.

Answer 11

That's why I mentioned the \(\dfrac{\text{Part}}{\text{Whole}}\) concept from the very beginning. Notice that the triangle is inside of the rectangle which means its area is part of the area of the rectangle.

Answer 12

Aha! Makes sense, thank you very much!