The square has 4 equal sides.  Let's assume a side is .

Each side of the square has a length of .

The area of the square is:

Substitute  into the area to solve for .

Since one side of the square is , substitute the value of   to determine the length of the square side.

The area of a square is:

Substitute the area and solve for the side.

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

The side length of the square is:

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

Eq. 2

Now we plug  into the first equation to find the value of :

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

Since there are four sides, .

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

Since  bisects  into two equal parts, this creates an isosceles triangle .

Therefore . The sum of the angles in a triangle is 180 degrees.

Therefore 

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,  and  are each bisected into two equal angles.

Therefore,  , which creates a triangle in the upper right quadrant of the kite. The sum of angles in a triangle is 180 degreees.

Thus,

Since  is only half of ,

In a rhombus, opposite angles are equal to each other. Therefore we can set  and  equal to one another and solve for :

Therefore,

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

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:

When we combine like terms, we get the following:

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

Now, we will divide both sides of the equation by 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: 

 or 

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

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

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:

We must then plug in our values of  and  given to us for the length and width.

So for this data:

Therefore, the amount of fencing needed to fully surround the dog's enclosure is .