Hello again, ejaykasai!

3. A certain city block is in the form of a parallelogram.

Two of it's sides measure 32 ft and 41 ft.

If the area of the block is 656 ft^2, what is the length if its longer diagonal? Code:

A *--------------------* D
/ /|
/ / |
32 / / |
/ 32/ | h
/ / |
/ / |
*--------------------*- - - *
B - - - - 41 - - - - C - - E

We have parallelogram $\displaystyle ABCD\!:\;\;AB = 32,\:BC = 41.$

Let the height be: $\displaystyle h \:=\:DE.$

Since $\displaystyle \text{(Area)} \:=\:\text{(base)} \times \text{(height)} $

. . we have: .$\displaystyle 41h \:=\:656 \quad\Rightarrow\quad h \:=\:16$

Code:

A *--------------------* D
/ * /|
/ * / |
32 / * / |
/ * 32/ | 16
/ * / |
/ * / |
*--------------------*- - - *
B - - - - 41 - - - - C - - E

In right triangle $\displaystyle DEC\!:\;CE^2 + 16^2 \:=\:32^2 \quad\Rightarrow\quad CE^2 = 768$

Hence: .$\displaystyle CE = 16\sqrt{3} \quad\Rightarrow\quad BE \:=\:41 + 16\sqrt{3}$

In right triangle $\displaystyle DEB\!:\;\;BD^2 \:=\:(41+16\sqrt{3})^2 + (16)^2$

. . $\displaystyle BD^2 \;=\;1681 + 1312\sqrt{3} + 768 + 256 \;=\;2705 + 1312\sqrt{3}$

Therefore: .$\displaystyle BD \;=\;\sqrt{2705 + 1312\sqrt{3}} \;\approx\;70.55\text{ ft}$