Area of a Triangle: Using ratios for calculation

Let's calculate the area of triangle ABC by following these steps:

• Step 1: Calculate AE using the ratio. Given that side BC is $\frac{1}{3}$ of AE, if we call AE = 6 (as shown or suggested by the illustration, considering AE is the vertical height), then $BC = \frac{1}{3} \times 6 = 2$.
• Step 2: The height of triangle ABC is given by the length AE, which is 6.
• Step 3: Calculate the area using the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times BC \times AE = \frac{1}{2} \times 2 \times 6 = 6$.

Therefore, the area of triangle ABC is $\text{6}$.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the length of side BC using the given ratio to AE.
• Step 2: Use the formula for the area of a triangle with the calculated BC and given AE.

Let's work through these steps:

Step 1: Calculate BC
Given that BC is $\frac{1}{4}$ the length of AE, and AE is 12, we find the length of BC as follows:

$BC = \frac{1}{4} \times 12 = 3$

Step 2: Use the formula for the area of a triangle
The formula is given by $A = \frac{1}{2} \times \text{base} \times \text{height}$.

In this context, BC serves as the base, and AE serves as the height. Thus, we compute the area:

$A = \frac{1}{2} \times BC \times AE = \frac{1}{2} \times 3 \times 12 = \frac{1}{2} \times 36 = 18$

Therefore, the area of triangle $\triangle$ ABC is $\mathbf{18}$.

Thus, the correct answer is choice 4: $\mathbf{18}$.

To solve the problem, we will follow these steps:

• Step 1: Using $AE = 10$, calculate $BC$ using the information that $BC = \frac{1}{5} \times AE$.
• Step 2: Apply the formula for the area of a triangle.

We proceed with the calculations:
Step 1: Since $BC = \frac{1}{5} \times 10 = 2$.
Step 2: Assume $AB$ is perpendicular to $BC$ to apply the formula $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.

The triangle $\triangle ABC$ is vertical, implying that base $BC$ and height $AE$ can be used because it suggests $AE$ is perpendicular to $BC$. Thus,
$\text{Area} = \frac{1}{2} \times AE \times BC = \frac{1}{2} \times 10 \times 2 = 10$.

Therefore, the area of the triangle is $10$.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the length of side BC using the given ratio.
• Step 2: Find the area of the triangle using the area formula.
• Step 3: Verify the solution with the provided choice options.

Now, let's work through each step:

Step 1: Side AE is provided as 9. Since side BC is $\frac{2}{3}$ of side AE, we calculate:

$BC = \frac{2}{3} \times AE = \frac{2}{3} \times 9 = 6$

Step 2: Because triangle ABC is involved in calculation, the area of triangle BCE becomes $\text{Area} = \frac{1}{2} \times BC \times height$, and assumed height perfectly completes these side lengths within the triangle grid. Given no specific height, the functional area equals set simplification calculations:

Substitute the values to find the area of triangle BCE:

$\text{Area of } \triangle BCE = \frac{1}{2} \times BC \times AE = \frac{1}{2} \times 6 \times 9$

Calculate:

$\text{Area} = \frac{1}{2} \times 54 = 27$

Therefore, the solution to the problem is $27$.

To solve this problem, let's proceed step-by-step:

Step 1: Calculate the length of $BC$
Given $BC = \frac{5}{8} \times AE$ and $AE = 16$, calculate:

$BC = \frac{5}{8} \times 16 = \frac{80}{8} = 10$

Step 2: Use the triangle area formula $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
Substitute $BC = 10$ as the base and $AE = 16$ as the height:

$\text{Area} = \frac{1}{2} \times 10 \times 16$

Step 3: Perform the calculation:

$\text{Area} = \frac{1}{2} \times 160 = 80$

Therefore, the area of triangle $\triangle ABC$ is $\boxed{80}$.

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

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 $x = 2$.