A parallelogram has sides with lengths of 24 and 9 . If the parallelogram's area is 135 , what is the length of its longest diagonal?

1 Answer
Nov 6, 2016

31.53

Explanation:

With these types of questions, it's better to draw a symbolic geometric shape, in this case, a parallelogram. After drawing the parallelogram, draw the two diagonals. You'll see that the longest diagonal is to the opposite side of the obtuse angle. After determining the longest diagonal, you can erase the other diagonal.

Now there should be two triangles in the parallelogram like this;
enter image source here

Consider b=24 , a=9 and c the diagonal we want to find.
This triangle has an area that is half of the parallelogram which is 135/2=67.5
In order to find c, we need to find the angle B. Using the sine area theorem, which is 1/2*a*b*sinB=A(ABC) in this triangle;

1/2*24*9*sinB=67.5

sinB=0.625

B~~141.32 degrees

Now we know the angle B, we can find c by using the cosine theorem, which would be c^2= a^2+b^2-2*a*b*cosB in this triangle.

c^2 = 24^2+9^2-2*24*9*cosB

c^2 ~~ 657 + 432*0.781

c^2 ~~ 994.2

c ~~ sqrt994.2 ~~ 31.53