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