# A parallelogram has sides 12cm and 18cm and a contained angle of 78 degrees. Find the shortest diagonal?

##### 1 Answer

The length of the shorter diagonal is

#### Explanation:

Let the sides of your parallelogram be

#a = 12 "cm"# and#b = 18 "cm"# .

and your angle be

#gamma = 78^@# , the angle between#a# and#b# (or#b# and#a# )

As two adjacent angles in a parallelogram are suplementary, we know that the adjacent angle to

#beta = 180^@ - gamma = 180^@ - 78^@ = 102^@#

Now, we can use the law of cosines to compute both diagonals in the parallelogram.

Let

The law of cosines states:

#d^2 = a^2 + b^2 - 2ab * cos(gamma)#

#e^2 = a^2 + b^2 - 2ab * cos(beta)#

Thus, we can compute the lengths of the diagonals as follows:

#d^2 = 12^2 + 18^2 - 2 * 12 * 18 * cos(78^@)#

# = 144 + 324 - 432 * cos(78^@)#

# ~~ 468 - 432 * 0.208#

# ~~ 378.18#

# => d ~~ 19.45 "cm"#

And for the other diagonal,

#e^2 = 12^2 + 18^2 - 2 * 12 * 18 * cos(102^@)#

# = 468 - 432 * (-0.208) #

# ~~ 557.82#

# => e ~~ 23.62 "cm"#

Thus, the length of the shorter diagonal is