Area of a Triangle: Using ratios for calculation

$$$$The area of trapezoid ABCD is X cm².

The line AE creates triangle AED and parallelogram ABCE.

The ratio between the area of triangle AED and the area of parallelogram ABCE is 1:3.

Calculate the ratio between sides DE and EC.

To calculate the ratio between the sides we will use the existing figure:

$\frac{A_{AED}}{A_{ABCE}}=\frac{1}{3}$

We calculate the ratio between the sides according to the formula to find the area and then replace the data.

We know that the area of triangle ADE is equal to:

$A_{ADE}=\frac{h\times DE}{2}$

We know that the area of the parallelogram is equal to:

$A_{ABCD}=h\times EC$

We replace the data in the formula given by the ratio between the areas:

$\frac{\frac{1}{2}h\times DE}{h\times EC}=\frac{1}{3}$

We solve by cross multiplying and obtain the formula:

$h\times EC=3(\frac{1}{2}h\times DE)$

We open the parentheses accordingly:

$h\times EC=1.5h\times DE$

We divide both sides by h:

$EC=\frac{1.5h\times DE}{h}$

We simplify to h:

$EC=1.5DE$

Therefore, the ratio between is: $\frac{EC}{DE}=\frac{1}{1.5}$

$1:1.5$

Since the side BC is $\frac{1}{5}$ side AE.

Calculate the area of the triangle:

10

Since the side BC is $\frac{1}{4}$ side AE.

Calculate the area of the triangle:

18

Since the side BC is $\frac{2}{3}$ side AE.

Calculate the area of the triangle:

27

Since the side BC is $\frac{1}{3}$ side AE.

Calculate the area of the triangle:

6

Since the side BC is $\frac{5}{8}$ side AE.

Calculate the area of the triangle:

80

The area of trapezoid ABCD

is 30 cm².

The line AE creates triangle AED and parallelogram ABCE.

The ratio between the area of triangle AED and the area of parallelogram ABCE is 1:2.

Calculate the ratio between sides DE and EC.

1

The area of the triangle below is equal to 10 cm² and its height is 5 times greater than its base.

Calculate X.

$x=2$