A parallelogram has sides with lengths of #16 # and #9 #. If the parallelogram's area is #81 #, what is the length of its longest diagonal?

Question

A parallelogram has sides with lengths of #16  # and #9  #.  If the parallelogram's area is #81  #, what is the length of its longest diagonal?

Theodore Abad · Accepted Answer

Approximately #23.98#.

Referring to the diagram, this solution boils down to three elemental steps:
1. Find length of #HB# using the area formula;
2. Find the length of #AH# by applying the Pythagoras theorem with knowledge about lengths of #AB# and #HB#.
3. Apply the Pythagoras theorem for one more time to find the length of #AC#, the longest diagonal.

Step One: Find #BH#, height on the longest side
#BH=\frac{Area}{AD}=\frac{81}{16}#

Step Two: Find #AH# and then #IC#
Apply the Pythagoras theorem in #RT\DeltaAHB#.
#AH=\sqrt{AB^2-HB^2}=sqrt{9^2-(\frac{81}{16})^2}=\frac{45\sqrt{7}}{16}#

#IB=AH# since #AIBH# is a rectangle. Therefore #IC=IB+BC=16+\frac{45\sqrt{7}}{16}#

Step Three: Find #AC#
Apply the Pythagoras theorem one more time in #RT\DeltaAIC#.
#AC=\sqrt{AI^2+IC^2}\approx23.98#

Sophia Alfred · Answer

Length of longer diagonal is #23.98# unit.

We know the area of the parallelogram as #A_p=s_1*s_2*sin theta or sin theta=81/(16*9)=0.5625# # :. theta=sin^-1(0.5625) ~~34.23^0 #.Consecutive angles are supplementary #:.theta_2=180-34.23~~145.77^0#. Longer diagonal can be found by applying cosine law: #d_l= sqrt(s_1^2+s_2^2-2*s_1*s_2*costheta_2)# #=sqrt(16^2+9^2-2*16*9*cos145.77) ~~ 23.98 unit # Length of longer diagonal is #23.98# unit [Ans]

Evelyn Arboleda · Answer

undefined To find the length of the longest diagonal of a parallelogram, you can use the formula:

Longest diagonal = √(a^2 + b^2 + 2ab)

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

Given:
a = 16
b = 9

Plugging in the values:
Longest diagonal = √(16^2 + 9^2 + 2 * 16 * 9)

Longest diagonal = √(256 + 81 + 288)

Longest diagonal = √(625)

Longest diagonal = 25

Therefore, the length of the longest diagonal of the parallelogram is 25 units.

Mason Adams · Answer

undefined To find the length of the longest diagonal of a parallelogram, you can use the formula:

$$ \text{Diagonal} = \sqrt{(side1)^2 + (side2)^2 + 2 \times side1 \times side2} $$

Given the sides of the parallelogram as 16 and 9, and the area as 81, we can use the formula for area of a parallelogram:

$$ \text{Area} = \text{base} \times \text{height} $$

$$ 81 = 16 \times \text{height} $$

$$ \text{height} = \frac{81}{16} $$

Now, substituting these values into the formula for the diagonal:

$$ \text{Diagonal} = \sqrt{16^2 + 9^2 + 2 \times 16 \times 9} $$

$$ \text{Diagonal} = \sqrt{256 + 81 + 288} $$

$$ \text{Diagonal} = \sqrt{625} $$

$$ \text{Diagonal} = 25 $$

So, the length of the longest diagonal of the parallelogram is 25.