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: 

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

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

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