A parallelogram has sides with lengths of #18 # and #5 #. If the parallelogram's area is #42 #, what is the length of its longest diagonal?

1 Answer
May 22, 2018

Longest diagonal #color(crimson)(d_1 = 22.47# units

Explanation:

Length of the longest diagonal is given by the formula

#d_1 = sqrt(a^2 + b^2 + (2 a b theta)# where

a is one pair of parallel lines, b the other pair and theta the angle between non-parallel sides.

Given : #a = 18, b = 5, A_p = 42#

#a * b * sin theta = Area = A_p#

#sin theta = 42 / (18 * 5) = 1/2#

#theta = sin ^-1 (1/2) = pi/6#

Longest diagonal #d_1 = sqrt(18^2 + 5^2 + (2 * 18 * 5 * cos (pi/3)# units

#color(crimson)(d_1 = 22.47# units

Shortest diagonal #d_2 = sqrt(a^2 + b^2 - (2 a b cos theta)#

#d_2 = sqrt(18^2 + 5^2 - ( 2*18*5*cos(pi/3)#

#d_2 = 13.9# units