The square has 4 equal sides.  Let's assume a side is $\displaystyle y$.

$\displaystyle 4y=12x$

$\displaystyle y=3x$

Each side of the square has a length of $\displaystyle 3x$.

The area of the square is:

$\displaystyle A=s^2= (3x)^2= 9x^2$

Substitute $\displaystyle 81x$ into the area to solve for $\displaystyle x$.

$\displaystyle 9x^2=81x$

$\displaystyle x=9$

Since one side of the square is $\displaystyle 3x$, substitute the value of $\displaystyle x=9$  to determine the length of the square side.

$\displaystyle y=3(9)=27$

The area of a square is:

$\displaystyle A=s^2$

Substitute the area and solve for the side.

$\displaystyle s^2 = 4x^2-16$

$\displaystyle s=\sqrt{4x^2-16} = \sqrt{4(x^2-4)} = \sqrt4 \sqrt{x^2-4}= 2\sqrt{x^2-4}$

The quantity inside the square root may not be factorized in attempt to eliminate the square root!

The side length of the square is:

$\displaystyle 2\sqrt{x^2-4}$

By definition, a rhombus is a quadrilateral with four equal sides whose angles do not all equal 90 degrees. To find the perimeter, we must find the values of x and y. In order to do so, we must set up a system of equations where we set two sides equal to each other. Any two sides can be used to create these systems.

Here is one example:

Eq. 1

$\displaystyle 2x+7=xy+1$

$\displaystyle 2x+6=xy$

$\displaystyle y=\frac{2x+6}{x}$

Eq. 2

$\displaystyle 2x+7=2(x+y)-3=2x+2y-3$

$\displaystyle 10=2y$

$\displaystyle y=5$ 

Now we plug $\displaystyle y=5$ into the first equation to find the value of $\displaystyle x$:

$\displaystyle 5=\frac{2x+6}{x}$

$\displaystyle 5x=2x+6$

$\displaystyle 3x=6$

$\displaystyle x=\frac{6}{3}=2$

Plugging these values into any of the three equations will give us the length of one side equaling 11.

Since there are four sides, $\displaystyle 11(4)=44$.

A rectangle has two sets of parallel sides with all angles equaling 90 degrees. 

Since $\displaystyle \angle ADB$ bisects $\displaystyle \overline{EC}$ into two equal parts, this creates an isosceles triangle $\displaystyle ABD$.

Therefore $\displaystyle \angle DAB = \angle DBA$. The sum of the angles in a triangle is 180 degrees.

Therefore $\displaystyle \angle ADB = 180-35-35 = 110$

A rhombus is a quadrilateral with two sets of parallel sides as well as equal opposite angles. Since the lines drawn inside the rhombus are diagonals, $\displaystyle \angle DAB,\angle ABC,\angle BCD,$ and $\displaystyle \angle CDA$ are each bisected into two equal angles.

Therefore, $\displaystyle \angle ADE=\angle ABE$ , which creates a triangle in the upper right quadrant of the kite. The sum of angles in a triangle is 180 degreees.

Thus,

$\displaystyle 180 = (x+35)+(2x+10)+90=3x+135$

$\displaystyle 3x=45$

$\displaystyle x=15$ 

$\displaystyle \angle EAB = 2(15)+10=40$

Since $\displaystyle \angle EAB$ is only half of $\displaystyle \angle DAB$,

$\displaystyle \angle DAB=2(40)=80$

$\displaystyle \angle DAB=\angle BCD=80$

In a rhombus, opposite angles are equal to each other. Therefore we can set $\displaystyle \angle A$ and $\displaystyle \angle C$ equal to one another and solve for $\displaystyle y$:

$\displaystyle 15+2y=\frac{y+96}{2}$

$\displaystyle 30+4y=y+96$

$\displaystyle 3y=66$

$\displaystyle y=\frac{66}{3}=22$

Therefore, $\displaystyle \angle A = \angle C=59$

A rhombus, like any other quadrilateral, has a sum of angles of 360 degrees.

$\displaystyle \angle B = \frac{360-59-59}{2}=121$ 

In order to solve this problem we need the following key piece of knowledge: the interior angles of a quadrilateral add up to 360 degrees. Now, we can write the following equation:

$\displaystyle 71+9x+5x+3x=360$

When we combine like terms, we get the following:

$\displaystyle 71+17x=360$

We will need to subtract 71 from both sides of the equation:

$\displaystyle 71-71+17x=360-71$

$\displaystyle 17x=289$

Now, we will divide both sides of the equation by 17.

$\displaystyle \frac{17x}{17}=\frac{289}{17}$

$\displaystyle x=17$

We now have a value for the x-variable; however, the problem is not finished. The question asks for the measure of the smallest angle. We know that the smallest angle will be one of the following: 

$\displaystyle 71^{\circ}$ or $\displaystyle 3x^{\circ}$

In order to find out, we will substitute 17 degrees for the x-variable.

$\displaystyle 3(17^{\circ})=51^{\circ}$

Because 51 degrees is less than 71 degrees, the measure of the smallest angle is the following:

$\displaystyle 51^{\circ}$

To answer this question, we must find the perimeter of the fence the homeowner is wanting to create.

To find the perimeter of a rectangle, we multiply the length by two, multiply the width by two, and add these two numbers together. The equation can be represented as this:

$\displaystyle 2L + 2W = Perimeter$

We must then plug in our values of $\displaystyle 12\:ft$ and $\displaystyle 14\:ft$ given to us for the length and width.

So for this data:

$\displaystyle 2L + 2W = 2(12) + 2(14) = 24 + 28 = 52$

Therefore, the amount of fencing needed to fully surround the dog's enclosure is $\displaystyle 52\:ft$.