Solve for x

Area of a rectangle A = lw = x(3x) = 3x² = 108

x² = 36

x = 6

Area of a rectangle is = lw

Perimeter = 2(l+w)

We are given 34 = 2(l+w) or 17 = (l+w)

possible combinations of l + w

are 1+16, 2+15, 3+14, 4+13... ect

We are also given the area of the rectangle is 52 meters squared.

Do any of the above combinations when multiplied together= 52 meters squared? yes 4x13 = 52

Therefore the longest side of the rectangle is 13 meters

To find the width, we must take half of the length, which means we must divide the length by 2. Then we must take 2 more than that number, which means we must add 2 to the number. Combining these, we get:

w = ½ l + 2

The width equals 3 times the length, so 3l, plus an additional two inches, so + 2, = 3l + 2

We are given the rectangle's diagonal \(\displaystyle \dpi{100} (d=25 in)\) and width \(\displaystyle (w=7in)\) and are asked to find its perimeter. The diagonal forms two right triangles within the rectangle; therefore, we can use the Pythagorean theorem to find the length of the rectangle's missing side. Then we can use the formula \(\displaystyle P=2l+2w\) to find the perimeter of the rectangle.

\(\displaystyle a^{2}+b^{2}=c^{2}\)

In our case, we can rename the variables to match our triangle.

\(\displaystyle w^{2}+l^{2}=d^{2}\)

\(\displaystyle \dpi{100} \dpi{100} (7in)^{2}+l^{2}=(25in)^{2}\)

\(\displaystyle l^{2}=625in^{2}-49in^{2}\)

\(\displaystyle \sqrt{l^{2}}=\sqrt{576in^{2}}\)

\(\displaystyle l=24in\)

Now we can calculate the perimeter.

\(\displaystyle P=2l+2w\)

\(\displaystyle P=(2*7in)+(2*24in)\)

\(\displaystyle P=62in\)

We can solve this by setting up a proportion and solving for x,the length of the shorter side of the house. If the drawing is scale and is 6 : 8, then the actual house is x : 64. Then we can cross multiply so that 384 = 8x. We then divide by 8 to get x = 48. 

To find the diagonal of a rectangle we must use the side lengths to create a 90 degree triangle with side lengths of 5, 8, and a hypotenuse which is equal to the diagonal.

If we have a right angle triangle and a value for two of the three side lengths, we use the Pythagorean Theorem to solve for the length of the third side.

The two side lengths that meet to form the right angle are labeled \(\displaystyle a\) and \(\displaystyle b\) which are interchangeable for each side length.

The long length connecting them is labeled \(\displaystyle c\) and is known as the hypotenuse.

The Pythagorean Theorem states 

Take 5 and 8 and plug them into the equation as \(\displaystyle a\) and \(\displaystyle b\) to yield \(\displaystyle 5^2+8^2=c^2\)

First square the numbers \(\displaystyle 25+64=c^2\)

After squaring the numbers add them together \(\displaystyle 89=c^2\)

Once you have the sum, square root both sides \(\displaystyle \sqrt{89}=\sqrt{c^2}\)

After calculating our answer for the diagonal is \(\displaystyle 9.4\).

The diagonal of the rectangle is equivalent to finding the length of the hypotenuse of a right triangle with sides 3 and 4. Using the Pythagorean Theorem:

\(\displaystyle 3^{2}+4^{2} = hypotenuse^{2}\)

\(\displaystyle 25 = hypotenuse^{2}\)

\(\displaystyle hypotenuse = 5\)

Therefore the diagonal of the rectangle is 5 feet.

The length and width of the rectangle are in a ratio of 3:4, so the sides can be written as 3x and 4x.

We also know the area, so we write an equation and solve for x:

(3x)(4x) = 12x² = 108.

x² = 9

x = 3

Now we can recalculate the length and the width:

length = 3x = 3(3) = 9 centimeters

width = 4x = 4(3) = 12 centimeters

Using the Pythagorean Theorem we can find the diagonal, c:

length² + width² = c²

9² + 12² = c² 

81 + 144 = c²

225 = c²

c = 15 centimeters

We have to be careful of our units. The floor is given in feet and the tile in inches. Since the floor is 6ft x 3ft. we can say it is 72in x 36in, because 12 inches equals 1 foot. If the tiles are 9in x 9in we can fit 8 tiles along the length and 4 tiles along the width. To find the total number of tiles we multiply 8 x 4 = 32. Alternately we could find the area of the floor (72 x 36, and divide by the area of the tile 9 x 9)