# Ratios of Area in Rectangles

• May 9th 2010, 06:22 PM
lyyy94
Ratios of Area in Rectangles
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.
• May 9th 2010, 06:49 PM
sa-ri-ga-ma
Quote:

Originally Posted by 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.

1) Area of ABCD = AB*BC
Area of AEC = 1/2*AE*BC
• May 9th 2010, 09:11 PM
Soroban
Hello, lyyy94!

Quote:

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

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 $x.$

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

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

Quote:

2. $M$ and $N$ are midpoints of sides $DA$ and $AB$ of square $ABCD.$
What is the ratio of the area of $\Delta AMN$ 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