A parallelogram has sides A, B, C, and D. Sides A and B have a length of #18 # and sides C and D have a length of # 9 #. If the angle between sides A and C is #pi/4 #, what is the area of the parallelogram?

Answer 1

Use a right triangle to find the height and eventually find the Area = #81sqrt(2)#

We start with a parallelogram with sides 18 and 9 and an angle between the sloping sides is #(pi/4)#. We're asked for the area of the figure.

The area of a parallelogram is:

#A=bh# where b is the base, or long side, and h is the perpendicular distance between the two long sides. To find it, we'll set up a triangle where a part of the base is one side, the height is the other side, and the sloping line (side C or D) is the hypotenuse.
There are 2 angles we know - the intersection of the height with the base, which is a right angle, and the angle between side C and the base, #(pi/4)#, or 45 degrees.
#(pi/4)# is a special angle in geometry - the relationship of the sides of a right triangle that has an angle of #(pi/4)# is #x, x, xsqrt(2)# where the first 2 x terms are the two sides and the #xsqrt(2)# is the hypotenuse.
We can, therefore, divide 9 by the #sqrt(2)# to find the height:
Height = #9/sqrt(2)#

We can multiply this by the long side (side A = 18) to arrive at:

#A=bh#
#A=18*9/sqrt(2)#

Let's clean this up by getting rid of the square root in the denominator and working the math:

#A=(18*9)/sqrt(2)*sqrt(2)/sqrt(2)=((18)(9)sqrt(2))/2=81sqrt(2)#
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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