A parallelogram has sides with lengths of #16 # and #8 #. If the parallelogram's area is #80 #, what is the length of its longest diagonal?

Question

A parallelogram has sides with lengths of #16  # and #8  #.  If the parallelogram's area is #80  #, what is the length of its longest diagonal?

Caroline Aucoin · Accepted Answer

longest diagonal #=22.8" "#units

From the given: two sides 16 and 8 and area=80, are enough to compute for the height h of the parallelogram. Area = 16h 80=16h #h=5# Compute the larger interior angle: Let that angle be X #X=180^@-sin^-1 (h/8)=180^@-38.6822^@=141.318^@# Now using Cosine Law to compute the longer diagonal #d# #d=sqrt(16^2+8^2-2*(16)*(8)*cos (141.318^@)# #d=22.8" "#units God bless...I hope the explanation is useful.

Aaliyah Anderson · Answer

undefined To find the length of the longest diagonal of the parallelogram, we can use the formula for the area of a parallelogram:

Area = base × height

Since the sides of the parallelogram are given as 16 and 8, and the area is given as 80, we can find the height of the parallelogram using the formula:

$80 = 16 \times \text{height}$

Solving for the height:

$\text{height} = \frac{80}{16} = 5$

Now, the longest diagonal of a parallelogram can be found using the Pythagorean theorem. In a parallelogram, the diagonals bisect each other and form right angles.

Let $d$ be the length of the longest diagonal. Then, according to the Pythagorean theorem:

$d^2 = (\text{side})^2 + (\text{height})^2$

Substituting the values:

$d^2 = 16^2 + 5^2$

$d^2 = 256 + 25$

$d^2 = 281$

Therefore, $d = \sqrt{281}$ or approximately $16.76$.