Area of a Parallelogram: Using variables

To solve this problem, we'll use the area formula for a parallelogram:

• Step 1: Identify and assign base and height. Assume $X$ is the base and the given side (3) is the height.
• Step 2: Apply the formula $S = b \times h$, where $b$ is base and $h$ is height.
• Step 3: Since $S = 21$, substitute it into the equation $21 = X \times 3$.
• Step 4: Solve for $X$.

Let's work through these steps:

Step 1: Assume $X$ is the base, and 3 is the height.

Step 2: Use the formula $S = b \times h = X \times 3$.

Step 3: Substitute $S = 21$:

Step 4: Solve for $X$:

Simplifying gives:

Therefore, the solution to the problem is $X = 7$.

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

• Step 1: Identify the given information.
• Step 2: Apply the appropriate formula for a parallelogram's area.
• Step 3: Rearrange and calculate the unknown $X$.

Now, let's work through each step:

Step 1: We have that the area $S$ is $45$ and the base $b$ is $5$.

Step 2: We use the formula for the area of a parallelogram $S = b \times h$, where in this case, $h$ is $X$. So, we have:

$45 = 5 \times X$

Step 3: Rearrange the equation to solve for $X$:

$X = \frac{45}{5}$

$X = 9$

Therefore, the length of side $X$ is $9$.

To determine the area of the parallelogram, we follow these steps:

• Step 1: Identify the given dimensions of the parallelogram.
• Step 2: Apply the formula for the area of a parallelogram.
• Step 3: Calculate using the given expressions for base and height.

Let's perform each step:

Step 1: From the problem, the base of the parallelogram is given as $3X$ and the height as $4Y$.

Step 2: Use the formula for the area $A$ of a parallelogram: $A = \text{base} \times \text{height}$.

Step 3: Plug these expressions into the formula:
$A = 3X \times 4Y$

Perform the multiplication:

$A = 3 \cdot 4 \cdot X \cdot Y = 12XY$

Thus, the area of the parallelogram is $12XY$.

When comparing this result to the given answer choices, the correct choice is:

• Choice 1: $12xy$

To find $X$, we use the formula for the area of a parallelogram:

$\text{Area} = \text{base} \times \text{height}$

Given that the area is 392 cm$^2$, and assuming $2X$ is the base and $4X$ is the height, we substitute these values into the formula:

$392 = (2X) \times (4X)$

Simplifying the right side gives us:

$392 = 2 \cdot 4 \cdot X^2 = 8X^2$

To solve for $X^2$, divide both sides by 8:

$X^2 = \frac{392}{8}$

$X^2 = 49$

Taking the square root of both sides, we find:

$X = \sqrt{49}$

$X = 7$

Therefore, the solution to the problem is $X = 7$ cm.

To find the side length $DC$ of parallelogram $ABCD$ with an area of $72y \, \text{cm}^2$:

• Given: Area = $72y \, \text{cm}^2$ and height = $9 \, \text{cm}$ (from the base).
• Formula for area: $\text{Area} = \text{base} \times \text{height}$.
• Rearrange to solve for base $DC$: $\text{base} = \frac{\text{Area}}{\text{height}}$.
• Substitute the given values: $\text{base} = \frac{72y}{9}$.
• Simplify: $\text{base} = 8y$.

Therefore, the length of $DC$ is $8y \, \text{cm}$.

To solve this problem, let's analyze and calculate step by step:

The formula for the area of a parallelogram is given by:

$\text{Area} = \text{Base} \times \text{Height}$

We're given:

• The area is $4x$.
• The height $AE = 2$.

We need to find the base $AD$. Let's plug these values into the formula:

$4x = AD \times 2$

Now, solve for $AD$ by dividing both sides by 2:

$AD = \frac{4x}{2} = 2x$

Therefore, the length of $AD$ is $2x$.

To find the area of the parallelogram ABCD, we use the following information and process:

• Step 1: Recognize that $AB$, which equals 20 units, is the base of the parallelogram.
• Step 2: The height corresponding to this base is the line segment from $A$ perpendicular to $CD$, denoted $AE = x$.
• Step 3: Use the formula for the area of a parallelogram: $\text{Area} = \text{base} \times \text{height}$.

Given:

• Base $AB = 20$.
• Height $AE = x$.

$\text{Area} = AB \times AE = 20 \times x = 20x$

Therefore, the area of the parallelogram ABCD is $20x$.

The correct answer is $\boxed{20x}$.

To solve this problem, we will follow these steps:

• Step 1: Setup and solve the equations for side lengths $AB$ and $AD$.
• Step 2: Calculate the area using the base $AD$ and the given height of 2 cm.

Let's begin:

Step 1: Calculate side lengths

Given that the perimeter is 22 cm, we have:

The equation simplifies to:

We are also given:

Substitute this in the first equation:

Now, substitute $AD = 8$ back into the expression for $AB$:

Step 2: Calculate the area

With $AD = 8$ cm as the base (since the problem specifies height to $AD$) and the given height of 2 cm, the area is calculated as:

Therefore, the area of the parallelogram is 16 cm².

To find the area of the parallelogram, we first use the given ratio of 4:1 between the height and side $AB$. This tells us that if side $AB$ is $2X$, then the height must be four times smaller, because we are considering the ratio in terms of the order given $\frac{\text{height}}{\text{AB}} = 4:1$.

Given side $AB = 2X$, the height of the parallelogram is:

$\text{height} = \frac{1}{4} \times 2X = \frac{1}{2}X$.

Now, we calculate the area of the parallelogram using the formula:

$\text{Area} = \text{base} \times \text{height}$.

Here, base = $2X$, and height = $\frac{1}{2}X$.

Thus,

$\text{Area} = 2X \times \frac{1}{2}X = X \times X = X^2$.

Therefore, the area of the parallelogram is $x^2$.

The perimeter of the parallelogram is calculated as follows:

$S_{ABCD}=AB+BC+CD+DA$ Since ABCD is a parallelogram, each pair of opposite sides is equal, and therefore, AB=DC and AD=BC

According to the figure that the side of the parallelogram is 2 times larger than the side adjacent to it, it can be argued that$AB=DC=2BC$

We inut the data we know in the formula to calculate the perimeter:

$P_{ABCD}=2BC+BC+2BC+BC$

We replace the given perimeter in the formula and add up all the BC coefficients accordingly:

$24=6BC$

We divide the two sections by 6

$24:6=6BC:6$

$BC=4$

We know that$AB=DC=2BC$We replace the data we obtained (BC=4)

$AB=DC=2\times4=8$

As ABCD is a parallelogram, then all pairs of opposite sides are equal, therefore BC=AD=4

To find EC we use the formula:$A_{ABCD}=AB\times EC$

We replace the existing data:

$24=8\times EC$

We divide the two sections by 8$24:8=8EC:8$

$3=EC$