The formula for the relationship between diagonals and sides of a parallologram is

 $\displaystyle (d_{1})^{2} + (d_{2})^{2} = 2a^{2} + 2b^{2}$,

where $\displaystyle d_{1}$ represents one diagonal, 

$\displaystyle d_{2}$ represents the other diagonal, 

$\displaystyle a$ represents a side, and 

$\displaystyle b$ represents the adjoining side.  

So, in this problem, substitute the known values and solve for the missing diagonal. 

$\displaystyle 14^{2} + (d_{2})^{2} = 2(12)^{2} + 2(5)^{2} \rightarrow 196 + d^{2} = 288 + 50$

$\displaystyle \rightarrow d^{2} = 142 \rightarrow d = \sqrt{142}$

So, the missing diagonal is $\displaystyle \sqrt{142}$ cm.

In a parallogram, diagonals bisect one another, thus you can set the two segments that are labeled in the picture equal to one another, then solve for $\displaystyle x$.  

So, 

$\displaystyle 2x - 2 = x + 12 \rightarrow x = 14$.

If $\displaystyle x = 14$, then you can substitute 14 into each labeled segment, to get a total of 52.

$\displaystyle 2(14)-2+14+12$

$\displaystyle 28-2+14+12=52$

In a parallelogram, the diagonals bisect one another, so you can set the labeled segments equal to each other and then solve for $\displaystyle x$.  

$\displaystyle x + 14 = 5x - 10 \rightarrow 24 = 4x \rightarrow 6 = x$.  

If $\displaystyle x = 6$, then you substitute 6 into each labeled segment, to get a total of 40.

$\displaystyle 6+14+5(6)-10$

$\displaystyle 20+30-10=40$

In a parallelogram, the diagonals bisect each other, so you can set the labeled segments equal to one another and then solve for $\displaystyle x$. 

$\displaystyle 20 - 3x = 2x - 4 \rightarrow 24 = 5x \rightarrow 4.8 = x$.  

Then, substitute 4.8 for $\displaystyle x$ in each labeled segment to get a total of 11.2 for the diagonal length.

$\displaystyle 20-3(4.8)+2(4.8)-4$

$\displaystyle 20-14.4+9.6-4=11.2$

Write the formula to find the side of the square given the area.

$\displaystyle A=s^2$

Find the side.

$\displaystyle 6=s^2$

$\displaystyle s=\sqrt6$

The diagonal of the square can be solved by using the Pythagorean Theorem.

$\displaystyle a^2+b^2=c^2$

$\displaystyle a=b=s=\sqrt6$

Substitute and solve for the diagonal, $\displaystyle c$.

$\displaystyle (\sqrt6)^2+(\sqrt6)^2 =c^2$

$\displaystyle 6+6=c^2$

$\displaystyle c^2=12$

$\displaystyle c=\sqrt{12}=2\sqrt3$

Write the diagonal formula for a square.

$\displaystyle d=s\sqrt2$

Substitute the side length and reduce.

$\displaystyle d=3a\sqrt2$

One characteristic of a rectangle is that its diagonals are congruent. Since the diagonals of Parallelogram $\displaystyle ABCD$ are of different lengths, it cannot be a rectangle.

One characteristic of a rhombus is that its diagonals are perpendicular; no restrictions exist as to their lengths. Whether or not the diagonals are perpendicular is not stated, so the figure may or may not be a rhombus.

In order to find the perimeter of this parallelogram, apply the formula: 
$\displaystyle \small Perimeter=2(base+side)$.

The solution is:

$\displaystyle \small P=2(8+5)$

$\displaystyle \small \small P=2(13)=26$

To find the perimeter of this parallelogram, first find the length of the side: $\displaystyle \small h\times\frac{1}{3}$.

Since, $\displaystyle \small h=48$, the side must be $\displaystyle \small 48\div 3=16$.

Then apply the formula: 

$\displaystyle \small p=2(base+side)$

$\displaystyle \small p=2(96+16)$

$\displaystyle \small p=2(112)=224$