A parallelogram has sides with lengths of #15 # and #12 #. If the parallelogram's area is #54 #, what is the length of its longest diagonal?

Question

A parallelogram has sides with lengths of #15  # and #12  #.  If the parallelogram's area is #54  #, what is the length of its longest diagonal?

Aurora Ayala · Accepted Answer

Length of its longest diagonal is #26.69#

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 #15# and #12# and area is #54# we have

#15xx12xxsintheta=54# or #sintheta=54/(15xx12)=0.3#

#costheta=sqrt(1-(0.3)^2)=sqrt(1-0.09)#

= #sqrt(0.91)=0.954#

Then larger diagonal of parallelogram would be given by

#sqrt(a^2+b^2+2abcosthet)=sqrt(15^2+12^2+2xx15xx12xx0.954#

= #sqrt(225+144+360xx0.954)=sqrt(369+343.44)#

= #sqrt712.44=26.69#

Sadie Adams · Answer

undefined To find the length of the longest diagonal of a parallelogram with sides 15 and 12 and an area of 54, you can use the formula for the area of a parallelogram: area = base × height. Rearrange this formula to solve for the height: height = area / base. Substitute the given values into the formula to find the height. Once you have the height, you can use the Pythagorean theorem to find the length of the longest diagonal. The diagonal's length is the square root of the sum of the squares of the two side lengths and twice the square of the height. Calculate the length of the longest diagonal using these steps.