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 length multiplied by width:

$S=AB\times AD$

Let's 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 all sides together

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

$AD=BC=4$

$AB=CD=5$

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

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

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

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

We'll group similar terms:

$30x-16=44$

$30x=44+16$

$30x=60$

$x=2$

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:

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

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:

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