A parallelogram has sides A, B, C, and D. Sides A and B have a length of #5 # and sides C and D have a length of # 8 #. If the angle between sides A and C is #(5 pi)/8 #, what is the area of the parallelogram?

Answer 1

#"Total Area" ~~ 36.96#

So let's sketch the scenario. Remember that in a parallelogram, opposite angles are congruent and adjacent angles are supplementary.

To find the area, we will split the shape into three shapes. Two right angled triangles and a rectangle.

We find the length of the vertical lines (the height of the triangles) by trigonometry. Remember SOH CAH TOA, so sine is opposite/hypotenuse.

#sin((3pi)/(8)) = y/8#

#y = 8sin((3pi)/8) ~~ 7.39# but we'll keep it in it's exact form for now.

We find the base of the triangles with the same technique:

#sin(pi/8) = x/8#

#x = 8sin(pi/8) ~~ 3.06#

The area of each triangle is given by

#A_("triangle") = 1/2a*b*sin(C)#

If we use #a and b# to be the base and height this reduces to

#A_("triangle") = 1/2xy = 32sin((3pi)/8)sin(pi/8) #

The area outside the rectangle is given by #2A_("triangle") ~~ 22.63 #.

The rectangle's dimensions are given by the height of the triangles and #5- "the base of the triangles"#

#A_("rectangle") = y(5-x) = 8sin((3pi)/8)*(5-8sin(pi/8)) ~~ 14.33#

#"Total Area" = A_("rectangle") + 2A_("triangle")#

#"Total Area" ~~ 36.96#

As a sense check, if the shape was just a rectangle the area would be 40 so this is in the right neck of the woods at least.

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Answer 2

To find the area of the parallelogram, you can use the formula:

[ \text{Area} = \text{base} \times \text{height} ]

where the base is the length of either side A or B, and the height is the perpendicular distance between sides A and C.

Given that the length of side A (and also side B) is 5, and the length of sides C (and also side D) is 8, we can use the formula:

[ \text{Area} = 5 \times 8 \times \sin\left(\frac{5\pi}{8}\right) ]

Solving this, the area of the parallelogram is approximately 22.57 square units.

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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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