Area of a Triangle: Using ratios for calculation

To solve this problem, we shall adhere to the following steps:

• Step 1: Utilize the area formula for triangles.
• Step 2: Simplify the equation to find the variable $x$.
• Step 3: Verify the result against the multiple-choice options.

Now, let us execute these steps:

Step 1: Start by applying the triangle area formula $A = \frac{1}{2} \times \text{base} \times \text{height}$.
The given area is $10 \, \text{cm}^2$, the base is $x$, and the height is $5x$. Thus, the formula becomes:

$10 = \frac{1}{2} \times x \times 5x$

Step 2: Simplify the equation:
$10 = \frac{1}{2} \times 5x^2$ $10 = \frac{5}{2}x^2$

Multiply both sides by $2$ to eliminate the fraction:

$20 = 5x^2$

Divide both sides by $5$:

$4 = x^2$

Take the square root of both sides:

$x = 2$

So, the value of $x$ is $\boxed{2}$.

Step 3: Upon reviewing the given multiple-choice options, the answer $x = 2$ corresponds to one of the listed choices, ensuring our calculations align with the expected solution.

Therefore, the solution to the problem is $x = 2$.

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: $\frac{EC}{DE}=\frac{1}{1.5}$