A parallelogram has sides with lengths of #14 # and #8 #. If the parallelogram's area is #24 #, what is the length of its longest diagonal?

Question

A parallelogram has sides with lengths of #14  # and #8  #.  If the parallelogram's area is #24 #, what is the length of its longest diagonal?

Sophia Alfred · Accepted Answer

Longest diagonal #color(green)(d_1 = 21.88# units

Longest diagonal #d_1 = a^2 + b^2 + 2 a b cos theta# #A_p = a b sin theta# #A_p = 24, a = 14, b = 8# #sin theta = 24 / (14 * 8) = 0.2143# #theta = sin^-1 0.2143 = 0.216# #cos theta = 0.9768# Longest diagonal #d_1 = sqrt(14^2 + 8^2 + (2 * 14 * 8 * 0.9768))# #color(green)(d_1 = 21.88# units Shortest diagonal #d_2 = sqrt(14^2 + 8^2 - (2 * 14 * 8 * 0.9768))# #color(crimson)(d_2 = 6.42# units

Aurora Ayala · Answer

undefined The length of the longest diagonal of the parallelogram can be found using the formula:

$$ \text{Longest diagonal} = \sqrt{a^2 + b^2 + 2ab\cos(\theta)} $$

Where $ a $ and $ b $ are the lengths of the sides of the parallelogram and $ \theta $ is the angle between them.

Given that the sides of the parallelogram are 14 and 8, and the area is 24, we can find the angle between the sides using the formula for the area of a parallelogram:

$$ \text{Area} = ab\sin(\theta) $$

Solving for $ \theta $, we get:

$$ \theta = \arcsin\left(\frac{\text{Area}}{ab}\right) $$

Then, we can use the formula for the longest diagonal to find its length.