Ratios of Area in Rectangles

May 2010
5
0
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.
 
Jun 2009
806
275
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
May 2006
12,028
6,341
Lexington, MA (USA)
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