A parallelogram has sides with lengths of #12 # and #8 #. If the parallelogram's area is #32 #, what is the length of its longest diagonal?

1 Answer
EZ as pi
Jun 11, 2016

#"diagonal" = 19.7#

Explanation:

In a parallelogram, the end of the longer diagonal extends beyond the end of the base, but by how much? Let's call it #x#.

The #height = "Area"/"base" rArr "h##= 32/12" = 8/3#

Consider the right-angled triangle at the side of the parallelogram:
It has sides of # x and 8/3# and the hypotenuse is 8.

#x^2 = 8^2 - (8/3)^2 = 512/9 " " rArr x = sqrt(512/9) = sqrt512/3#

The length of the base produced is therefore # (12 + sqrt512/3)#

Working in the new right-angled triangle, the length of the longer diagonal is now the hypotenuse of the triangle with sides of
# (12 + sqrt512/3) and 8/3#

Using Pythagoras: #"diagonal"^2 = (12 +sqrt512/3)^2 + (8/3)^2#
#"diagonal"^2 = 389.019#
#"diagonal" = 19.7#

Would working with surds rather than resorting to decimals be easier?

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