A parallelogram has sides with lengths of #14 # and #8 #. If the parallelogram's area is #49 #, 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 #49 #, what is the length of its longest diagonal?

Sadie Adams · Accepted Answer

Find the height, then use Pythagoras twice - first to find the length of the extended base, then to find the length of the longer diagonal.
Diagonal = #sqrt461.425 = 21.48# Draw a sketch of the parallelogram first! Find the height of the parallelogram. A = b x h #h = A/b = 49/14 = 7/2 = 3.5# The endpoint of the top line of the parallelogram is vertically above the base extended. We need to calculate the length of the extended line, let's call it #x#. Working in the right-angled triangle outside the parallelogram with Hypotenuse = 8 and height = 3.5 , gives the following: #x^2 = 8^2 -3.5^2 = 51.75# #x = sqrt51.75 " "# leave it in this form Length of the extended base = #14 + sqrt51.75~~21.194# Now work with the right-angled triangle with the longer diagonal# (d)# as the hypotenuse, and the two shorter sides being #3.5 and (14+sqrt51.75)# #d^2 = (14+sqrt51.75)^2 +3.5^2 = 461.425....# #d = 21.48#

Carson Arnold · Answer

undefined To find the length of the longest diagonal of a parallelogram with given side lengths, you can use the formula:

$$ \text{Area} = \text{base} \times \text{height} $$

Given that the area of the parallelogram is 49 and one of the sides (base) is 14, you can find the height using the formula:

$$ \text{Height} = \frac{\text{Area}}{\text{Base}} $$

$$ \text{Height} = \frac{49}{14} = 3.5 $$

Now, using the Pythagorean theorem, you can find the length of the longest diagonal (which is the hypotenuse of a right triangle formed by the two sides and the height):

$$ \text{Longest diagonal} = \sqrt{(\text{Base})^2 + (\text{Height})^2} $$

$$ \text{Longest diagonal} = \sqrt{(14)^2 + (3.5)^2} $$

$$ \text{Longest diagonal} = \sqrt{196 + 12.25} $$

$$ \text{Longest diagonal} = \sqrt{208.25} $$

$$ \text{Longest diagonal} \approx 14.43 $$

So, the length of the longest diagonal of the parallelogram is approximately 14.43 units.