A parallelogram has sides with lengths of #24 # and #9 #. If the parallelogram's area is #144 #, 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 #144 #, what is the length of its longest diagonal?

Allison Allen · Accepted Answer

Length of its longest diagonal is #31.29#

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 #144# we have

#24xx9xxsintheta=144# or #sintheta=144/(24xx9)=2/3#

#costheta=sqrt(1-(2/3)^2)=sqrt(1-4/9)#

= #sqrt(5/9)=1/3sqrt5=2.23607/3=0.7454#

Then larger diagonal of parallelogram would be given by

#sqrt(a^2+b^2-2abcostheta)=sqrt(24^2+9^2+2xx24xx9xx0.7454#

= #sqrt(576+81+432xx0.7454)=sqrt(657+322.0128)#

= #sqrt979.0128=31.29#

Noah Atkinson · Answer

undefined The length of the longest diagonal of the parallelogram can be found using the formula:

Diagonal^2 = Side1^2 + Side2^2 + 2 * Side1 * Side2 * cos(angle between them)

Given that the sides of the parallelogram are 24 and 9, and the area is 144, we can find the angle between them using the area formula:

Area = Side1 * Side2 * sin(angle between them)

After finding the angle, substitute it into the diagonal formula to find the length of the longest diagonal.

Theodore Abad · Answer

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

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

Given that the area of the parallelogram is 144 and one of its sides (base) is 24, we can rearrange the formula to solve for the height:

$$ \text{Height} = \frac{\text{Area}}{\text{Base}} $$

$$ \text{Height} = \frac{144}{24} = 6 $$

Now, since the height of a parallelogram is perpendicular to its base, it forms a right triangle with the longest diagonal (the hypotenuse) and half of the other side (the base). Using the Pythagorean theorem, we can find the length of the longest diagonal:

$$ \text{Longest diagonal}^2 = \text{base}^2 + \text{height}^2 $$

$$ \text{Longest diagonal}^2 = 24^2 + 6^2 = 576 + 36 = 612 $$

$$ \text{Longest diagonal} = \sqrt{612} \approx 24.7 $$

Therefore, the length of the longest diagonal of the parallelogram is approximately 24.7 units.