To answer this question, we must find the diagonal of a rectangle that is \(\displaystyle 8\: ft\) by \(\displaystyle 6\:ft\). Because a rectangle is made up of right angles, the diagonal of a rectangle creates a right triangle with two of the sides. 

Because a right triangle is formed by the diagonal, we can use the Pythagorean Theorem, which is:

\(\displaystyle a^{2}+b^{2}=c^{2}\)

\(\displaystyle a\) and \(\displaystyle b\) each represent a different leg of the triangle and \(\displaystyle c\) represents the length of the hypotenuse, which in this case is the same as the diagonal length. 

We can then plug in our known values and solve for \(\displaystyle c^{2}\)

\(\displaystyle a^{2}+b^{2}=c^{2}\rightarrow 8^{2}+6^{2}=c^{2}\rightarrow 64+36=c^{2}\rightarrow 100=c^{2}\)

We now must take the square root of each side so that we can solve for \(\displaystyle c\)

\(\displaystyle \sqrt{100}=\sqrt{c^{2}}\rightarrow 10=c\)

Therefore, the diagonal of the rectangle is \(\displaystyle 10\:ft\).

To find the length of the diagonal, we can consider only the triangle \(\displaystyle ABD\) and use the law of cosines to find the length of the unknown side.

The Law of Cosines:

\(\displaystyle c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )\)

Where \(\displaystyle c\) is the length of the unknown side, \(\displaystyle a\) and \(\displaystyle b\) are the lengths of the known sides, and \(\displaystyle C\) is the angle between \(\displaystyle a\) and \(\displaystyle b\). 

From the problem:

\(\displaystyle \overline{BD}^2 = \overline{AB}^2 + \overline{DA}^2 - 2\left ( \overline{AB} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle A\right )\)

\(\displaystyle \overline{BD}^2 = 35^2 + 82 ^2 - 2\cdot35\cdot82\cdot \cos \left ( 37^{\circ}\right )\)

\(\displaystyle \overline{BD}^2 = 3365\)

\(\displaystyle \overline{BD} = 58\)

To find the length of the diagonal, we can consider only the triangle \(\displaystyle ACD\) and use the law of cosines to find the length of the unknown side.

The Law of Cosines:

\(\displaystyle c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )\)

Where \(\displaystyle c\) is the length of the unknown side, \(\displaystyle a\) and \(\displaystyle b\) are the lengths of the known sides, and \(\displaystyle C\) is the angle between \(\displaystyle a\) and \(\displaystyle b\). 

From the problem:

\(\displaystyle \overline{AC}^2 = \overline{CD}^2 + \overline{DA}^2 - 2\left ( \overline{CD} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle D\right )\)

\(\displaystyle \overline{AC}^2 = 35^2 + 82 ^2 - 2\cdot35\cdot82\cdot \cos \left ( 143^{\circ}\right )\)

\(\displaystyle \overline{AC}^2 = 12553\)

\(\displaystyle \overline{AC} = 112\)

To find the length of the diagonal, we can consider only the triangle \(\displaystyle ACD\) and use the law of cosines to find the length of the unknown side.

The Law of Cosines:

\(\displaystyle c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )\)

Where \(\displaystyle c\) is the length of the unknown side, \(\displaystyle a\) and \(\displaystyle b\) are the lengths of the known sides, and \(\displaystyle C\) is the angle between \(\displaystyle a\) and \(\displaystyle b\). 

From the problem:

\(\displaystyle \overline{AC}^2 = \overline{CD}^2 + \overline{DA}^2 - 2\left ( \overline{CD} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle D\right )\)

\(\displaystyle \overline{AC}^2 = 12^2 + 40 ^2 - 2\cdot12\cdot40\cdot \cos \left ( 135^{\circ}\right )\)

\(\displaystyle \overline{AC}^2 = 2423\)

\(\displaystyle \overline{AC} = 49.2\)

To find the length of the diagonal, we can consider only the triangle \(\displaystyle ACD\) and use the law of cosines to find the length of the unknown side.

The Law of Cosines:

\(\displaystyle c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )\)

Where \(\displaystyle c\) is the length of the unknown side, \(\displaystyle a\) and \(\displaystyle b\) are the lengths of the known sides, and \(\displaystyle C\) is the angle between \(\displaystyle a\) and \(\displaystyle b\). 

From the problem:

\(\displaystyle \overline{AC}^2 = \overline{CD}^2 + \overline{DA}^2 - 2\left ( \overline{CD} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle D\right )\)

\(\displaystyle \overline{AC}^2 = 10^2 + 20 ^2 - 2\cdot10\cdot20\cdot \cos \left ( 120^{\circ}\right )\)

\(\displaystyle \overline{AC}^2 = 700\)

\(\displaystyle \overline{AC} = 26.5\)

To find the length of the diagonal, we can consider only the triangle \(\displaystyle ABD\) and use the law of cosines to find the length of the unknown side.

The Law of Cosines:

\(\displaystyle c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )\)

Where \(\displaystyle c\) is the length of the unknown side, \(\displaystyle a\) and \(\displaystyle b\) are the lengths of the known sides, and \(\displaystyle C\) is the angle between \(\displaystyle a\) and \(\displaystyle b\). 

From the problem:

\(\displaystyle \overline{BD}^2 = \overline{AB}^2 + \overline{DA}^2 - 2\left ( \overline{AB} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle A\right )\)

\(\displaystyle \overline{BD}^2 = 12^2 + 40 ^2 - 2\cdot12\cdot40\cdot \cos \left ( 45^{\circ}\right )\)

\(\displaystyle \overline{BD}^2 = 1065\)

\(\displaystyle \overline{BD} = 32.6\)

To find the length of the diagonal, we can consider only the triangle \(\displaystyle ABD\) and use the law of cosines to find the length of the unknown side.

The Law of Cosines:

\(\displaystyle c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )\)

Where \(\displaystyle c\) is the length of the unknown side, \(\displaystyle a\) and \(\displaystyle b\) are the lengths of the known sides, and \(\displaystyle C\) is the angle between \(\displaystyle a\) and \(\displaystyle b\). 

From the problem:

\(\displaystyle \overline{BD}^2 = \overline{AB}^2 + \overline{DA}^2 - 2\left ( \overline{AB} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle A\right )\)

\(\displaystyle \overline{BD}^2 = 10^2 + 20 ^2 - 2\cdot10\cdot20\cdot \cos \left ( 60^{\circ}\right )\)

\(\displaystyle \overline{BD}^2 = 300\)

\(\displaystyle \overline{BD} = \sqrt{300}\)