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A parallelogram has sides with lengths of #16 # and #15 #. If the parallelogram's area is #64 #, what is the length of its longest diagonal?

Answer 1

Maverick Smith

The longest diagonal #~~ 30.7#

Here is a reference to the properties of a Parallelogram

Let a = the length of the first side = 15

Let b = the length of the base = 16

The area of a parallelogram is:

#A = bh#

Substitute 64 for the area and 16 for the base:

#64 = 16h#

h = 4

We can use the equation #h = asin(pi - theta)# to find the sine of the angle between the base and the other side.

#4 = 15sin(theta)#

#sin(theta) = 4/15#

Use a well known trigonometric idenity to find the cosine:

#cos(theta) = sqrt(1 - sin^2(theta))#

#cos(theta) = sqrt(1 - (4/15)^2)#

#cos(theta) = sqrt(209)/15#

Because the two angles must add to #pi#, the cosine of the other angle is:

#cos(pi - theta) = cos(pi)cos(theta) + sin(pi)sin(theta)#

Simplify using #cos(pi) = -1 and sin(pi) = 0#

#cos(pi - theta) = -sqrt(209)/15#

The longest diagonal, c, can be found using the law of cosines

#c^2 = a^2 + b^2 - 2abcos(pi - theta)#

#c^2 = 15^2 + 16^2 - 2(16)(15)(-sqrt(209)/15)#

#c^2 = 225 + 256 + 32sqrt(209)#

#c ~~ 30.7#

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