All of the lengths with one mark have length 5, and all of the side lengths with two marks have length 4. With this knowledge, we can add side lengths together to find that one diagonal is the hypotenuse to this right triangle:

Using Pythagorean Theorem gives:

take the square root of each side

Similarly, the other diagonal can be found with this right triangle:

Once again using Pythagorean Theorem gives an answer of

To find the length of the diagonals, split the top side into 3 sections as shown below:

The two congruent sections plus 8 adds to 14. , so the two congruent sections add to 6. They must each be 3. This means that the top of the right triangle with the diagonal as a hypotenuse must be 11, since .

We can solve for the diagonal, now pictured, using Pythagorean Theorem:

take the square root of both sides

To solve for the area of the rhombus , we must use the equation , where  and  are the diagonals of the rhombus. Since the perimeter of the rhombus is , and by definition all 4 sides of a rhombus have the same length, we know that the length of each side is . We can find the length of the other diagonal if we recognize that the two diagonals combined with a side edge form a right triangle. The length of the hypotenuse is , and each leg of the triangle is equal to one-half of each diagonal. We can therefore set up an equation involving Pythagorean's Theorem as follows:

, where  is equal to one-half the length of the unknown diagonal.

We can therefore solve for  as follows:

 is therefore equal to 8, and our other diagonal is 16. We can now use both diagonals to solve for the area of the rhombus:

The area of rhombus  is therefore equal to 

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle  to find the length of diagonal . From the problem, we are given that the sides are  and . Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle  to find the length of diagonal . From the problem, we are given that the sides are  and . Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle  to find the length of side . From the problem, we are given  and . Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

The perimeter of the regular hexagon is 80, so each side measures one sixth of this, or 20. Also, since  is the midpoint of , . 

Also, each interior angle of a regular hexagon measures .

Below is the hexagon in question, with  indicated and  constructed; all relevant measures are marked. 

A triangle  is formed with , , and included angle measure . The length of the remaining side can be calculated using the Law of Cosines: 

where  and  are the lengths of two sides,  is the measure of their included angle, and  is the length of the third side. 

Setting , and , substitute and evaluate :

;

Taking the square root of both sides:

,

the correct choice.

The perimeter of the regular pentagon is 80, so each side measures one fifth of this, or 16. Also, since  is the midpoint of , . 

Also, each interior angle of a regular pentagon measures .

Below is the pentagon in question, with  indicated and  constructed; all relevant measures are marked. 

A triangle  is formed with , , and included angle measure . The length of the remaining side can be calculated using the law of cosines: 

where  and  are the lengths of two sides,  is the measure of their included angle, and  is the length of the third side. 

Setting , and , substitute and evaluate :

;

Taking the square root of both sides:

,

the correct choice.

Step 1: Recall the formula used to find the missing side(s) of a right triangle...

Step 2: Identify the legs and the hypotenuse in the formula...

 are the legs, and  is the hypotenuse.

Step 3: Plug in the values of a and b given in the question...

Step 3: A special rule about all triangles...

For any triangle, the measurements of any of the sides CANNOT BE zero.

So, the missing side is , or 

The perimeter of the regular pentagon is 80, so each side measures one fifth of this, or 16. Since  is the midpoint of , . 

Similarly, .

Also, each interior angle of a regular pentagon measures .

Below is the pentagon with the midpoints  and , and with  constructed. Note that perpendiculars have been drawn to  from  and , with feet at points  and , respectively.

 is a rectangle, so .

, or 

. Substituting:

For the same reason, 

.

Adding the segment lengths:

, 

Rond answer to the nearest tenth.