# Ratios of Area in Rectangles

#### lyyy94

These two problems are giving me a lot of trouble. If any one could give me some help in the right direction I'd really appreciate it.

1 In rectangle ABCD, point E is on side AB so that AE = 10 and EB = 5. What fraction of the area of the rectangle is inside triangle AEC?

2 M and N are the midpoints of consecutive sides of a square ABCD with vertex A in between M and N. What is the ratio of the area of triangle AMN to the area of the complete square.

Thanks again.

#### sa-ri-ga-ma

These two problems are giving me a lot of trouble. If any one could give me some help in the right direction I'd really appreciate it.

1 In rectangle ABCD, point E is on side AB so that AE = 10 and EB = 5. What fraction of the area of the rectangle is inside triangle AEC?

2 M and N are the midpoints of consecutive sides of a square ABCD with vertex A in between M and N. What is the ratio of the area of triangle AMN to the area of the complete square.

Thanks again.
1) Area of ABCD = AB*BC
Area of AEC = 1/2*AE*BC

#### Soroban

MHF Hall of Honor
Hello, lyyy94!

1. In rectangle $$\displaystyle ABCD$$, point $$\displaystyle E$$ is on side $$\displaystyle AB$$ so that: .$$\displaystyle AE = 10,\;\;EB = 5$$
What fraction of the area of the rectangle is inside triangle $$\displaystyle AEC$$?
Code:
      :      10       E   5   :
A o - - - - - - - o - - - o B
|  *:::::::::::::*      |
|     *:::::::::::*     |
|        *:::::::::*    |
x |           *:::::::*   | x
|              *:::::*  |
|                 *:::* |
|                    *:*|
D o - - - - - - - - - - - o C
15
The length of the rectangle is 15.
The width of the rectangle is $$\displaystyle x.$$

The area of $$\displaystyle \Delta AEC\:=\:\tfrac{1}{2}(10)(x) \:=\:5x$$
The area of rectangle $$\displaystyle ABCD \:=\:15x$$

The fraction (ratio) is: .$$\displaystyle \frac{5x}{15x} \;=\;\frac{1}{3}$$

2. $$\displaystyle M$$ and $$\displaystyle N$$ are midpoints of sides $$\displaystyle DA$$ and $$\displaystyle AB$$ of square $$\displaystyle ABCD.$$
What is the ratio of the area of $$\displaystyle \Delta AMN$$ to the area of the square?
Make a sketch and the answer is obvious . . .
Code:
              N
A o - - - o - - - o B
|:::::* | *     |
|:::*   |   *   |
|:*     |     * |
M o - - - + - - - *
| *     |     * |
|   *   |   *   |
|     * | *     |
D o - - - * - - - o C