A parallelogram has sides with lengths of #21 # and #12 #. If the parallelogram's area is #24 #, what is the length of its longest diagonal?

Question

A parallelogram has sides with lengths of #21  # and #12  #.  If the parallelogram's area is #24 #, what is the length of its longest diagonal?

Savannah Aguilar · Accepted Answer

The longest diagonal is #=33#

We have a parallelogram #ABCD#

#AD=12#

#AB=21#

Area = base x height

#24=21*h#

#h=24/21=8/7#

The distance between the base of the height and the corner #D# is

#=sqrt(AD^2-h^2)=sqrt(144-64/49)=sqrt(6992/49)=11.95#

Therefore,

#DB^2=h^2+(DC+11.95)^2#

#DB^2=64/49+(21+11.95)^2#

#DB^2=64/49+1085.7#

#DB^2=1087#

#DB=sqrt1087#

#DB=33#

Aaliyah Anderson · Answer

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

Longest diagonal = √(a^2 + b^2 + 2ab)

Where 'a' and 'b' are the lengths of the sides of the parallelogram.

Given that the sides have lengths of 21 and 12, and the area of the parallelogram is 24, we can use the formula for the area of a parallelogram:

Area = base * height

24 = 21 * height

height = 24 / 21 = 8/7

Now, using the Pythagorean theorem, we can find the length of the longest diagonal:

Longest diagonal = √(21^2 + (8/7)^2 + 2 * 21 * (8/7))

Longest diagonal ≈ √(441 + 64/49 + 48) ≈ √(441 + 64/49 + 48) ≈ √(441 + 128/49)

Longest diagonal ≈ √(441 + 128/49) ≈ √(8939/49) ≈ √(8939)/7

Longest diagonal ≈ √(8939)/7 ≈ 11.86 units