If sides A and B of a triangle have lengths of 2 and 4 respectively, and the angle between them is #(pi)/4#, then what is the area of the triangle?

1 Answer
Alan P.
Dec 13, 2015

#2sqrt(2)#

Explanation:

One way to do this is strictly geometrical.

Consider the square containing the given triangle with the side of length #4# as the diagonal.
We would have the structure below:
enter image source here
We are asked for the area of #triangleT#
and it is clearly
#color(white)("XXX")#the area of the #square#
#color(white)("XXX")#minus
#color(white)("XXX")#(the area of #triangleA# plus #triangleB#)

With a diagonal of length #4# each side of the square has a length of #2sqrt(2)# (as shown) based on the Pythagorean Theorem.
Therefore the area of the #square# is #2sqrt(2) xx 2sqrt(2) = 8#

The area of #triangleA# is (#1/2bh#)
#color(white)("XXX")1/2xx2sqrt(2)xx2= 4#

The area of #triangleB# is
#color(white)("XXX")1/2xx(2sqrt(2)-2)xx2sqrt(2) = 4-2sqrt(2)#

The area of #triangleT# is
#color(white)("XXX")8- (4+4-2sqrt(2)) = 2sqrt(2)#

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