A parallelogram has sides with lengths of #12 # and #8 #. If the parallelogram's area is #64 #, what is the length of its longest diagonal?

Question

A parallelogram has sides with lengths of #12  # and #8  #.  If the parallelogram's area is #64  #, what is the length of its longest diagonal?

Claire Akin · Accepted Answer

longest diagonal =12.00053677

If we split the parallelogram into using the smaller diagonal, the area would be 32 # cm^2# using Area=#1/2 xxa xxb xxsin C# #32=1/2xx8xx12xxsin theta# #sin theta=32/48=2/3# #theta=sin^(-1) [2/3]# #theta=41.8^@# Therefore the other angle is #180-41.8=138.2^@# Using the cosine rule to find the longest diagonal #a^2=b^2+c^2-sxxbxxcxxcosA# #a^2=12^2+8^2-12xx8xxsin 138.2^@# #a^2=208-96sin138.2^@# #a^2=144.0128829# a=12.00053677

Carson Arnold · Answer

undefined To find the length of the longest diagonal of a parallelogram with sides of 12 and 8 and an area of 64, we can use the formula for the area of a parallelogram, which is given as base multiplied by height.

Let's denote the base of the parallelogram as 12 and the height as h. We can set up the equation:

Area = base × height
64 = 12 × h

Now, solve for h:

h = 64 / 12
h = 5.33 (rounded to two decimal places)

Now that we have the height, we can use the Pythagorean theorem to find the length of the longest diagonal. The diagonal, in a parallelogram, forms a right triangle with the base and height.

Using the Pythagorean theorem (a^2 + b^2 = c^2), where a and b are the base and height of the parallelogram respectively, and c is the length of the diagonal, we have:

a = 12 (base)
b = 5.33 (height)
c = diagonal (which we want to find)

Substitute the values into the Pythagorean theorem:

12^2 + 5.33^2 = c^2
144 + 28.39 = c^2
172.39 = c^2

Now, solve for c:

c = √172.39
c ≈ 13.13

Therefore, the length of the longest diagonal of the parallelogram is approximately 13.13 units.