Perimeter of a Rectangle: Using variables

Since in a rectangle every pair of opposite sides are equal, we can claim that:

$AD=BC=x+2$

$AB=CD=x+3$

Since the perimeter of the rectangle is equal to 26, we can substitute the data into the formula:

$26=x+2+x+3+x+2+x+3$

$26=4x+10$

$26-10=4x$

$16=4x$

Let's divide both sides by 4:

$4=x$

Since in a rectangle every pair of opposite sides are equal, we can claim that:

$AD=BC=2x+1$

$AB=CD=4x+3$

Since the perimeter of the rectangle is equal to 20, we can substitute the data into the formula:

$20=2x+1+4x+3+2x+1+4x+3$

$20=12x+8$

$20-8=12x$

$12=12x$

Let's divide both sides by 12:

$1=x$

The area of the rectangle equals its length multiplied by its width:

$S=AB\times AD$

Let's first substitute the data into the formula:

$20=4\times(x+1)$

$20=4x+4$

$20-4=4x$

$16=4x$

$4=x$

Now we can calculate side AB:

$4+1=5$

The perimeter of the rectangle equals the sum of its sides.

Since each pair of opposite sides are equal in a rectangle, we can calculate that:

$AD=BC=4$

$AB=CD=5$

Finally, let's add all the sides together to find the perimeter:

$4+5+4+5=8+10=18$

To solve this problem, let's clearly follow these steps:

• Step 1: Identify the information given and needed for solving.

• Step 2: Apply the perimeter formula for a rectangle.

• Step 3: Solve for the unknown variable $x$.

Step 1: The rectangle has a perimeter $P = 32$. One pair of opposite sides is $2x$ and the other pair is 10.

Step 2: The perimeter of a rectangle is calculated by
$P = 2(l + w)$ where $l$ is the length and $w$ is the width.
Here, $l = 2x$ and $w = 10$.

Step 3: Substitute the given values into the formula:

$32 = 2(2x + 10)$

Expand the equation:

$32 = 4x + 20$

To solve for $x$, subtract 20 from both sides:

$32 - 20 = 4x$

$12 = 4x$

Finally, divide both sides by 4 to find $x$:

$x = \frac{12}{4}$

$x = 3$

Therefore, the solution to the problem is $x = 3$.

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

• Step 1: Identify the given information
• Step 2: Use the perimeter formula
• Step 3: Solve for $x$

Now, let's work through each step:

Step 1: The problem gives us that the perimeter of the rectangle is 28, one side (length) is 11, and the other side (width) is $x$.
Step 2: We'll use the formula for the perimeter of a rectangle: $P = 2(l + w)$. In this problem, $l = 11$ and $w = x$ so:

$28 = 2(11 + x)$

Step 3: Divide both sides by 2 to isolate the expression inside the parentheses:

$14 = 11 + x$

Step 4: Subtract 11 from both sides to solve for $x$:

$x = 14 - 11 = 3$

Therefore, the value of $x$ is $3$.

To solve this problem, we will determine $x$ using the perimeter formula for a rectangle. The steps are as follows:

• Step 1: Set up the perimeter equation for this problem using the formula $P = 2(l + w)$.
• Step 2: Identify the given dimensions: $l = 2x$ and $w = x$.
• Step 3: Substitute these dimensions into the perimeter equation: $2(2x + x) = 12$.
• Step 4: Simplify the equation to solve for $x$.

Now, let's follow these steps:

Step 1: The perimeter $P = 12$ units. Thus, we use the equation:

Step 2: Substitute the known values of length and width:

Step 3: Simplify the equation inside the parentheses:

Step 4: Divide both sides by 2 to solve for $x$:

Finally, divide by 3:

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

To find $x$, we'll use the concept of the rectangle's perimeter.

Step 1: Identify the given expressions for length and width.

• Length $L = 4 + 2x$
• Width $W = x - 3$
• Perimeter $P = 8$

Step 2: Use the formula for the perimeter of a rectangle: $P = 2(L + W)$.

Substitute the given expressions into the formula:

Step 3: Simplify the equation within the parentheses:

Step 4: Distribute the 2:

Step 5: Solve for $x$:

Therefore, the value of $x$ is 1.

To solve this problem, we'll use the perimeter of a rectangle formula:

• Step 1: Identify the side lengths.
• Step 2: Use the perimeter formula to set up the equation.
• Step 3: Solve for $x$.

Now, let's work through each step:

Step 1: The side lengths are $x - 10$ and $4x - 8$.
Step 2: The perimeter $P$ is given by $2(x - 10) + 2(4x - 8) = 24$.

Simplify the equation:

Combine like terms:

Step 3: Solve for $x$:

Therefore, the value of $x$ is $\boxed{6}$.

Area of triangle ADB:

$\frac{AD\times AB}{2}$

Let's list the known data:

$9=\frac{x+2\times6}{2}$

$9=(x+2)\times3$

$9=3x+6$

$9-6=3x$

$3=3x$

$1=x$

Side AD equals:

$1+2=3$

Since in a rectangle, each pair of opposite sides are equal, we can state that:

$AD=BC=3$

$AB=CD=6$

Now we can calculate the perimeter of the rectangle:

$3+6+3+6=6+12=18$

Given that the perimeter of triangle BCD is 20

We can therefore insert the existing data and calculate as follows:

$20=10+6+x+2$

$20=16+x+2$

$20-16-2=x$

$x=2$

Now we can calculate the BC side: 2+2=4

Perimeter of the rectangle ABCD:

$6+6+4+4=12+8=20$

To solve the problem, we'll begin by setting up the equation for the perimeter of rectangle $AEFD$:

The perimeter $P$ of a rectangle is given by the formula:

We're told that the perimeter of rectangle $AEFD$ is 30. The length is $2x + 3$ and the width is $2 - x$. Thus, the perimeter equation is:

Let's simplify the expression inside the parentheses:

So the equation becomes:

Now, distribute the 2:

Subtract 10 from both sides of the equation:

Divide both sides by 2 to solve for $x$:

Therefore, the value of $x$ that satisfies the given perimeter is $x = 2$.

To solve the problem of finding $x$ given the perimeter of rectangle ABCD, we follow these steps:

• Step 1: The problem states that the perimeter of rectangle ABCD is 48. The expressions related to $x$ represent sides of this rectangle.
• Step 2: Identify the sides using expressions in the diagram:
  • The complete top and bottom sides (width) are divided into $3-x$ and full vertical is $5+x$.
  • One vertical height is $8+2x$.
• Step 3: Use the perimeter formula for a rectangle: $P = 2(\text{width} + \text{height})$.
• Step 4: Substitute in the formula and solve for $x$.

Now, let's apply these steps:

Express the perimeter using the given: $2((3-x) + (5+x)) + 2(8+2x) = 48$.

Simplify the equation:

$2(3-x + 5+x) + 2(8+2x) = 48$

$2(8) + 2(8+2x) = 48$

$16 + 16 + 4x = 48$

Combine like terms:

$32 + 4x = 48$

Isolate $4x$:

$4x = 48 - 32$

$4x = 16$

Solve for $x$:

$x = \frac{16}{4}$

$x = 4$

Therefore, the value of $x$ is $x = 4$.

Let's calculate the perimeter of rectangle GDEF using the given data:

$10x-8+10x-8+5x+5x=44$

We'll group similar terms:

$30x-16=44$

$30x=44+16$

$30x=60$

$x=2$

To solve this problem, we'll need to find the value of $X$ and then use this value to find the perimeter of the rectangle. Follow these detailed steps:

• Step 1: Set up the equation for the area.
  Given the area of the rectangle is 7, and the sides are $(X + 5)$ and $(X - 1)$, the equation becomes:

$(X + 5)(X - 1) = 7$
Expanding the left side, we have: $X^2 + 5X - X - 5 = 7$ $X^2 + 4X - 5 = 7$

• Step 2: Simplify and solve the quadratic equation.
  By moving all terms to one side, we have:

$X^2 + 4X - 5 - 7 = 0$ $X^2 + 4X - 12 = 0$

• Step 3: Factor the quadratic equation.
  We factor the equation as:

$(X + 6)(X - 2) = 0$

• Step 4: Solve for $X$ using the zero-product property.
  The solutions for $X$ are:

$X + 6 = 0 \Rightarrow X = -6$ $X - 2 = 0 \Rightarrow X = 2$

• Step 5: Use the positive $X$ value to find the perimeter.
  Since dimensions cannot be negative, $X = 2$. Thus, the rectangle’s dimensions become:

$\text{Length} = X + 5 = 2 + 5 = 7$
$\text{Width} = X - 1 = 2 - 1 = 1$

• Step 6: Calculate the perimeter using the dimensions.

$P = 2 \times (\text{Length} + \text{Width}) = 2 \times (7 + 1) = 2 \times 8 = 16$

Therefore, the perimeter of the rectangle is 16.

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

• Step 1: Write the expression for the perimeter of rectangle GCHF using the given variables.
• Step 2: Set up the equation by equating the expression to 18.
• Step 3: Solve the equation for $x$.

Let's work through each step:

Step 1: The perimeter of rectangle GCHF is given by the formula:

$\text{Perimeter} = 2 \times (\text{length}) + 2 \times (\text{width})$

According to the diagram, the dimensions of the rectangle GCHF are:

- GH = $3 + x$ (length),

- GF = $5x$ (width).

Thus, the perimeter formula becomes:

$2 \times (3 + x) + 2 \times (5x)$

Step 2: Set the equation equal to the given perimeter, 18:

$2(3 + x) + 2(5x) = 18$

Step 3: Simplify and solve for $x$:

Distribute the 2:

$6 + 2x + 10x = 18$

Combine like terms:

$6 + 12x = 18$

Subtract 6 from both sides:

$12x = 12$

Divide both sides by 12 to isolate $x$:

$x = 1$

Therefore, the solution to the problem is $x = 1$.

After squaring all sides, we can calculate the area as follows:

$4^2=16$

Since we are given that the area of the square equals the area of the rectangle , we will write an equation with an unknown since we are only given one side length of the parallelogram:

$16=1\times x$

$x=16$

In other words, we now know that the length and width of the rectangle are 16 and 1, and we can calculate the perimeter of the rectangle as follows:

$1+16+1+16=32+2=34$