A parallelogram has sides with lengths of #16 # 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 #16  # and #9  #.  If the parallelogram's area is #45  #, what is the length of its longest diagonal?

Wyatt Anderson · Accepted Answer

<mathjax>#24.71# <img src="https://tutor.hix.ai 
As shown in the diagram, <mathjax>#ABCD#</mathjax> is the parallelogram. 
Area of parallelogram <mathjax>#A=bxxH#</mathjax> 
Given <mathjax>#b=9 and A=45#</mathjax>, 
<mathjax>#=> H=45/9=5#</mathjax> 
<mathjax>#DeltaBCE =#</mathjax> right triangle 
<mathjax>#=> BC^2=BE^2+CE^2#</mathjax>, <mathjax>#(BC=16, BE=H=5)#</mathjax> 
<mathjax>#=> CE=sqrt(16^2-5^2)=15.2#</mathjax> 
<mathjax>#=> DE=DC+CE=9+15.2=24.2#
<mathjax>#DeltaBDE#</mathjax> is also a right triangle 
<mathjax>#BD#</mathjax> is the hypotenuse and also the longer diagonal of the parallelogram. 
<mathjax>#=> BD^2=BE^2+DE^2#</mathjax> 
<mathjax>#=> BD=sqrt(5^2+24.2^2)=24.71#
Hence, the length of the longer diagonal =<mathjax>#BD=24.71#

Evelyn Arboleda · Answer

undefined The length of the longest diagonal of the parallelogram is 17.