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

Question

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

Sadie Adams · Accepted Answer

Length of its longest diagonal is #32.96#

Area of a parallelogram is given by #axxbxxsintheta#,

where #a# and #b# are two sides of a parallelogram and #theta# is the angle included between them.

As sides are #24# and #9# and area is #24# we have

#24xx9xxsintheta=24# or #sintheta=24/(24xx9)=1/9#

#costheta=sqrt(1-(1/9)^2)=sqrt(1-1/81)#

= #sqrt(80/81)=1/9sqrt80=8.9443/9=0.9938#

Then larger diagonal of parallelogram would be given by

#sqrt(a^2+b^2-2abcostheta)=sqrt(24^2+9^2+2xx24xx9xx0.9938#

= #sqrt(576+81+432xx0.9938)=sqrt(657+429.3216)#

= #sqrt1086.3216=32.96#

Autumn Armstrong · Answer

undefined The length of the longest diagonal of the parallelogram is 10.

Carson Arnold · Answer

undefined To find the length of the longest diagonal of the parallelogram, we can use the formula for the area of a parallelogram:

Area = base × height

Given that the area is 24 and one side (base) is 24, we can find the height using the formula:

24 = 24 × height

Solving for height:

height = 24 ÷ 24 
height = 1

Now, since the height is 1, and we know the other side (9) represents the length of the diagonal, we can use the Pythagorean theorem to find the length of the longest diagonal:

Diagonal² = (Side₁)² + (Side₂)²
Diagonal² = 9² + 1²
Diagonal² = 81 + 1
Diagonal² = 82

Taking the square root of both sides to find the length of the diagonal:

Diagonal = √82

Therefore, the length of the longest diagonal of the parallelogram is √82 units.