A parallelogram has sides with lengths of #5 # and #9 #. If the parallelogram's area is #45 #, what is the length of its longest diagonal?

Question

A parallelogram has sides with lengths of #5  # and #9  #.  If the parallelogram's area is #45  #, what is the length of its longest diagonal?

Wyatt Anderson · Accepted Answer

Diagonal length: #sqrt(106)~~10.3# units

With sides of #5# and #9# and and area of #45# sq. units the parallelogram is actually a rectangle. #color(white)("XXX")#To see this, notice that using #9# as the base #color(white)("XXX")#with an area of #45#, the perpendicular height #color(white)("XXX")#must be #5# units. The length of the diagonal of a rectangle can be calculated based on the length of its sides, using the Pythagorean Theorem. In this case: #color(white)("XXX")"diagonal length "=sqrt(5^2+9^2)# #color(white)("XXXXXXXXXXXXX")=sqrt(25+81)# #color(white)("XXXXXXXXXXXXX")=sqrt(106)#

Violet Aldrich · Answer

I am advised that the found condition is mathematically compatible with the shape being a rectangle.

#L=sqrt(106) larr# Exact answer

#L.~~10.296# to 3 decimal places#larr# Approximate answer

When in doubt do a very quick and rough sketch in the margin. For this question it should take about 5 to 6 seconds

#color(blue)("Comment")#
Notice that #5xx9=45# which is the same as the given area. Thus the parallelogram is also a rectangle. Thus you can go straight in and use Pythagoras. #L=sqrt(5^2+9^2)#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Beginnings of a mathematical approach if angle "GAB<90^o)#

#color(blue)("Method")#
#color(brown)("The longest diagonal is AD")#

Determine the length of DE (h) using area
Determine the length of FE and hence AE
Solve for AD using Pythagoras
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine h")#

If you were to get a pair of scissors and cut out triangle ABG you will discover that it will fit exactly over the empty space DEF. The result being a rectangle.

The area of the rectangle is:
#9" units"xxh" units"->9h = 45" units"^2#

Divide both sides by #color(red)(9)#

#color(green)(9h=45color(white)("ddd")->color(white)("ddd")9/color(red)(9)h=45/color(red)(9) )#

But #9/9=1# giving:

#color(green)(color(white)("ddddddddd")->color(white)("ddddd")h=45/9#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Determine FE")#

Now we find out that #h=45/9=5# which is the same length as the slope. Consequently FE=0 and #color(magenta)("The shape is a rectangle")#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Greatest slope length is the diagonal of the rectangle.

Let the slope length be #L#

#L=sqrt(9^2+5^2)= sqrt(106)=sqrt(2xx56)#

Both 2 and 56 are prime numbers so we can not simplify any further

#L=sqrt(106)# Exact answer

#L=10.29563....~~10.296# to 3 decimal places

James Atkins · Answer

undefined To find the length of the longest diagonal of the parallelogram, you can use the formula:

$ \text{Area} = \text{Base} \times \text{Height} $

Given that the area is 45 and one side is 5, you can solve for the height:

$ 45 = 5 \times \text{Height} $

$ \text{Height} = \frac{45}{5} $

$ \text{Height} = 9 $

Now, you can use the Pythagorean theorem to find the length of the longest diagonal:

$ \text{Diagonal}^2 = 5^2 + 9^2 $

$ \text{Diagonal}^2 = 25 + 81 $

$ \text{Diagonal}^2 = 106 $

$ \text{Diagonal} = \sqrt{106} $

So, the length of the longest diagonal is approximately $ \sqrt{106} $ units.