A parallelogram has sides with lengths of #14 # and #8 #. If the parallelogram's area is #84 #, 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 #84 #, what is the length of its longest diagonal?

Aurora Ayala · Accepted Answer

The longest diagonal #=20.203 # The parallelogram with sides #a,b# has an area computed as #A = a times b times sin(theta) = 84# Solving for #theta# we obtain #theta_0 = 2.29353#[rad] Associating now an oriented segment to each side we get #vec a = a{cos(theta_0),sin(theta_0)}, vec b = b{1,0}# The two diagonals are obtained as #vec d_1 = vec a + vec b# #vec d_2 = vec a - vec b# Computing their length we get #norm(vec d_1) = 10.5753# #norm(vec d_2) = 20.203#

Parker Allen · Answer

undefined To find the length of the longest diagonal of the parallelogram, you can use the formula for the area of a parallelogram, which is given by $ \text{Area} = \text{base} \times \text{height} $. In this case, the base is 14 and the height is the length of the longest diagonal. So, we have $ 84 = 14 \times \text{height} $. Solving for the height, we get $ \text{height} = \frac{84}{14} = 6 $.

Now, in a parallelogram, the length of the longest diagonal is the same as the length of the shorter side opposite to it. Therefore, the length of the longest diagonal is 8 units.