Perimeter of a Parallelogram: Using variables

Since in a parallelogram each pair of opposite sides are equal and parallel:

$BC=AD=2x$

$AB=CD=x$

Now let's substitute the known data into the formula for calculating the perimeter:

$80=2x\times2+2\times x$

$80=4x+2x$

$80=6x$

Let's divide both terms by 6:

$\frac{80}{6}=\frac{6x}{6}$

$\frac{80}{6}=x$

Let's simplify the fraction by 2

$\frac{40}{3}=x$

To find the perimeter of the parallelogram ABCD, we will use the formula for the perimeter of a parallelogram:

• The formula for the perimeter $P$ is $P = 2 \times (\text{length of one base} + \text{length of one side})$.

Given that:

• The length of the base (either AB or CD) is $X$.
• The length of the side (either AD or BC) is 10.

Substituting these values into the formula for the perimeter, we get:

Distribute the 2:

Therefore, the perimeter of parallelogram ABCD is $2X + 20$.

The correct answer, as per the choices given, is choice 4: $2x+20$.

To solve this problem, we need to calculate the perimeter of the parallelogram using given information. Here are the steps to find the solution:

• Step 1: Identify the Longest Side.
  The longest side of the parallelogram, denoted by $X$, is given as $a$. Therefore, $a = X$.
• Step 2: Determine the Other Side Length.
  Given $a > 2b$, typically $X = 2b$ as a common interpretation for solving problems.
  Thus, the other side $b$ is half the longer side: $b = \frac{X}{2}$.
• Step 3: Apply the Perimeter Formula.
  The perimeter $P$ of a parallelogram is calculated as $P = 2(a + b)$.
  Plug in the values: $a = X$, $b = \frac{X}{2}$.
  Thus, perimeter $P = 2\left(X + \frac{X}{2}\right) = 2\left(\frac{2X}{2} + \frac{X}{2}\right) = 2\left(\frac{3X}{2}\right) = 3X$.

Therefore, the perimeter of the parallelogram in terms of $X$ is $\mathbf{3X}$.

In a parallelogram, each pair of opposite sides are equal and parallel: AB = CD and AC = BD.

Given that the length of one side is 4 times greater than the other side equal to X, we know that:

$AB=CD=4AC=4BD$

Now we replace the data in this equation with out own (assuming that AB = CD = X):

$x=x=4AC=4BD$

We divide by 4:

$\frac{x}{4}=\frac{x}{4}=AC=BD$

Now we calculate the perimeter of the parallelogram and express both AC and BD using X:

$P=x+\frac{x}{4}+x+\frac{x}{4}$

$P=2x+\frac{x}{4}+\frac{x}{4}=2\frac{1}{2}x$

To solve the problem, follow these steps:

• Step 1: Identify the given side lengths.
  The longer side of the parallelogram is given as $0.5X$. The other side, being half the length of the long side, is $0.25X$.
• Step 2: Use the perimeter formula.
  The formula for the perimeter $P$ of a parallelogram is $P = 2a + 2b$, where $a$ and $b$ are the lengths of the sides.
• Step 3: Plug in the side lengths into the formula.
  Substitute $a = 0.5X$ and $b = 0.25X$ into the perimeter formula:
  $P = 2(0.5X) + 2(0.25X) = X + 0.5X = 1.5X$.

Thus, the perimeter of the parallelogram expressed in terms of $X$ is $1.5X$.

Therefore, the correct answer is choice $2$, which is $1.5X$.

To solve the problem, we'll apply the perimeter formula for a parallelogram. We are given that one side $a = X$ and the other side $b = X + 4$. The perimeter $P$ of a parallelogram is calculated by the formula:

$P = 2(a + b)$

Substitute the values of $a$ and $b$:
$a = X$
$b = X + 4$

Plug these into the formula:

$P = 2(X + (X + 4))$

Simplify the expression inside the parentheses:

$P = 2(2X + 4)$

Distribute the 2:

$P = 4X + 8$

Therefore, the perimeter of the parallelogram in terms of $X$ is $4X + 8$.

As is true of a parallelogram each pair of opposite sides are equal to one another

$AB=CD,AC=BD$

Given that AB > AC

Let's call AC by the name X and therefore:

$AB=2AC=2\times x=2x$

Now we know that:

$AB=CD=2x,AC=BD=x$

The perimeter is equal to the sum of all the sides together:

$2x+x+2x+x=6x$

As is true for a parallelogram each pair of opposite sides are equal:

$AB=CD=6,AC=BD=x$

Calculate X according to the given perimeter:

$20=6+6+x+x$

$20=12+2x$

$20-12=2x$

$8=2x$

$x=4$

The problem involves finding the value of $X$ in a parallelogram with sides given and a specified perimeter. We will use the formula for the perimeter of a parallelogram.

The formula for the perimeter $P$ of a parallelogram is:

Given that:

• One side $AB = a = 8$.
• The other side $AC = b = X + 2$.
• The perimeter $P = 30$.

Substitute the given values into the perimeter formula:

Simplify the expression inside the parentheses:

Now the equation becomes:

Divide both sides by 2:

Subtract 10 from both sides to solve for $X$:

Thus:

The value of $X$ is therefore $\textbf{5}$.

The problem involves calculating $X$ for a parallelogram with given side lengths and perimeter. Let's proceed step-by-step:

Step 1: First, recognize that in a parallelogram, opposite sides are equal:
- $AB = CD = 4$ (given)
- $AC = BD = X-2$

Step 2: Use the perimeter formula for the parallelogram:
$P = 2(a + b)$
where $a = AB = 4$ and $b = AC = X-2$.

Step 3: Plug the perimeter value and side lengths into the formula:
$10 = 2(4 + (X - 2))$

Step 4: Simplify and solve for $X$:
$10 = 2(4 + X - 2)$
$10 = 2(X + 2)$

Step 5: Divide both sides by 2 to eliminate the factor:
$5 = X + 2$

Step 6: Subtract 2 from both sides to isolate $X$:
$X = 3$

Therefore, the correct value of $X$ is $3$.

The corresponding choice is option 4.

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

• Step 1: Identify the given information: $AB = 10$, $AC = X$, and the perimeter is 30.
• Step 2: Apply the formula for the perimeter of a parallelogram: $2(a + b) = \text{perimeter}$.
• Step 3: Substitute the values and solve for $X$.

Now, let's work through each step:
Step 1: The problem provides us $AB = 10$, $AC = X$, and the perimeter as 30.
Step 2: The perimeter $P$ of a parallelogram with sides $AB$ and $AC$ is given by $P = 2(AB + AC)$.
Substitute the known values: $2(10 + X) = 30$.
Step 3: Simplify this equation: $2(10 + X) = 30$ Divide both sides by 2: $10 + X = 15$ Subtract 10 from both sides to solve for $X$: $X = 5$

Therefore, the solution to the problem is $X = 5$.

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

• Step 1: Use the formula for the perimeter of a parallelogram.
• Step 2: Set up the equation using the given information.
• Step 3: Solve for $X$ and find the length of AC.

Now, let's work through each step:
Step 1: Recall that the perimeter of a parallelogram is given by the formula $P = 2(a + b)$, where $a$ and $b$ are the lengths of adjacent sides.
Step 2: In our parallelogram, opposite sides are equal. Therefore, the perimeter formula can be expressed as: $P = 2(AB + AC) = 20$ Substituting the given lengths: $2(5 + 2X) = 20$
Step 3: Simplify the equation: $2 \times (5 + 2X) = 20 \\ 10 + 4X = 20 \\ 4X = 10 \\ X = 2.5$ Now, substitute back to find the length of side AC: $AC = 2X = 2 \times 2.5 = 5$

Therefore, the length of side AC is $\bold{5}$.

The correct choice from the given options is $\bold{3}$: $\bold{5}$.

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

• Step 1: Identify the formula for the perimeter of a parallelogram.
• Step 2: Substitute the given values into the perimeter formula.
• Step 3: Solve for the unknown variable $X$.
• Step 4: Determine the specific length of AC using the value of $X$.

Now, let's work through each step:

Step 1: The formula for the perimeter of a parallelogram is given by

where $a$ and $b$ are the lengths of the two pairs of sides.

Step 2: Substitute the known values into the formula. Here, let $a = AB = 7$ and $b = AC = 0.5X$.

Perimeter $= 21$, so:

Step 3: Solve for the unknown variable $X$.

First, divide both sides of the equation by 2 to isolate the terms inside the parenthesis:

Subtract 7 from both sides:

Multiply both sides by 2 to isolate $X$:

Step 4: Determine the length of AC:

Substitute back to find $0.5X = AC$:

Therefore, the length of side AC is $3.5$.

To solve this problem, we begin by using the formula for the perimeter of a parallelogram:

The perimeter $P$ is given by: $P = 2(a + b)$

Given: $P = 30 \text{ cm}$, and $CD = 2x$. We know $AB = 2x$ since opposite sides of a parallelogram are equal. So, we write:

• $P = 2(BC + CD) = 30$
• $2(BC + 2x) = 30$
• $BC + 2x = 15$ (after dividing both sides by 2)
• $BC = 15 - 2x$

Thus, the length of side $BC$ is given by:

$BC = 15 - 2x$

Therefore, the correct option is:

• Choice 1: $15 - 2x$

This matches the problem's given correct answer.

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.

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$24:8=8EC:8$

$3=EC$