Two opposite sides of a parallelogram have lengths of 8 . If one corner of the parallelogram has an angle of pi/8 and the parallelogram's area is 12 , how long are the other two sides?

2 Answers
Apr 7, 2017

Nasty answer. Here's a walkthrough.

Explanation:

Let h be the (perpendicular) height extending from one of the bases that measures length b = 8 to the other base having the same length.

since A = bh = 8h is the area of the parallelogram, we have

8h = 12
h = 12/8 = 3/2 is its height.

Let c be the length of one of the unknown sides.
From basic trigonometry,

sin(pi/8) = h/c.

We may obtain the value of sin(pi/8) from the half-angle formula for sine.

That value is
sin(pi/8) = sqrt(2 -sqrt(2))/2

So that...

sqrt(2 -sqrt(2))/2 = (3/2)/c

Cross multiply:

csqrt(2 -sqrt(2)) = 3

c = 3/(sqrt(2 -sqrt(2)))

If we demand that the expression be rationalized, then
multiply first by
sqrt(2 -sqrt(2))/sqrt(2 -sqrt(2)) and next by
(2 + sqrt(2))/(2 +sqrt(2)).

Apr 7, 2017

Picture

Explanation:

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If you also have the formula

A = bcsin(alpha)

where alpha is the angle included between sides b and c, you may get there more quickly.

12 = 8csin(pi/8).

Use the value of the sine, and it comes out quickly.