A parallelogram has sides with lengths of #15 # and #8 #. If the parallelogram's area is #12 #, what is the length of its longest diagonal?

1 Answer
Alan P.
Oct 31, 2016

Length of longest diagonal #22.974# (approx.)

Explanation:

Using a side of length #8# as the base,
since the area is #12#
the height relative to a side of length #8# is #color(purple)(h=12/8=3/2)#
enter image source here

The required diagonal (#color(red)(d)#) is
the hypotenuse of a right triangle formed by extending the base by an amount #color(purple)(x)# until the line segment terminates at a point which is at a right angle under the furthest upper vertex of the parallelogram.

The length of this extension (#color(purple)(x)#) can be calculated using the Pythagorean Theorem:
#color(white)("XXX")color(purple)x=sqrt(15^2-h^2)~~14.928# (using a calculator)

The requested diagonal now can also be calculated using the Pythagorean Theorem as:
#color(white)("XXX")color(red)d=sqrt((8+x)^2+h^2) ~~22.974# (again with a calculator)

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