A parallelogram has sides with lengths of #5 # and #9 #. If the parallelogram's area is #36 #, what is the length of its longest diagonal?

Question

A parallelogram has sides with lengths of #5  # and #9  #.  If the parallelogram's area is #36  #, what is the length of its longest diagonal?

Wyatt Anderson · Accepted Answer

Length of the longest diagonal AC #= color (purple)(12.6491)#

Given #l = 9, w = 5, Area = 36#

Area of the parallelogram = l * h = 36
#:. BE = CF = h = (Area)/l = 36 / 9 = 4#

AE = DF = a = sqrt(w^2-h^2) = sqrt(5^2 - 4^2) = 3#

AF = l + a = 9 + 3 = 12#

Longest diagonal AC #= sqrt(AF^2 + CF^2) = sqrt(12^2 + 4^2) = 12.6491#

James Atkins · Answer

undefined To find the length of the longest diagonal of the parallelogram, you can use the formula for the area of a parallelogram: Area = base * height. Since the area is given as 36 and the base is 9, you can find the height using the formula. Once you have the height, you can use the Pythagorean theorem to find the length of the longest diagonal, which is the hypotenuse of a right triangle formed by the sides of the parallelogram and its height.

The height can be calculated as follows: height = area / base = 36 / 9 = 4.

Now, you can use the Pythagorean theorem to find the length of the longest diagonal. Let's denote the sides of the parallelogram as a and b, and the diagonal as d. Then, using the theorem:

d² = a² + b²

Given that a = 9, b = 5, and h = 4, we can calculate:

d² = 9² + 4² = 81 + 16 = 97

So, d = √97 ≈ 9.85 units.

Therefore, the length of the longest diagonal of the parallelogram is approximately 9.85 units.