Two opposite sides of a parallelogram have lengths of #12 #. If one corner of the parallelogram has an angle of #pi/8 # and the parallelogram's area is #32 #, how long are the other two sides?

Question

Two opposite sides of a parallelogram have lengths of #12  #.  If one corner of the parallelogram has an angle of #pi/8   # and the parallelogram's area is #32      #, how long are the other two sides?

Violet Aldrich · Accepted Answer

#~=1# Area of Parallelogram #=32=ab sintheta# where #a=12# and #b=?# and #theta=(pi)/8# or #32=12timesbtimessin((pi)/8)# or #b=32/12times(0.383)# or #b~=1#

Autumn Armstrong · Answer

undefined The area $A$ of a parallelogram can be calculated using the formula $A = b \times h$, where $b$ is the length of the base and $h$ is the height. Given that one angle of the parallelogram is $\frac{\pi}{8}$, we can find the height $h$ using the sine function, since $h = b \sin(\theta)$, where $\theta$ is the given angle and $b$ is one side of length 12.

The area of the parallelogram is given as 32, and the length of the base $b$ is 12. Thus, we have:

$$32 = 12 \times h$$

To find $h$, we recognize that $h = 12 \sin\left(\frac{\pi}{8}\right)$. So, we need to solve for $h$ directly in terms of the area equation:

$$32 = 12 \times 12 \sin\left(\frac{\pi}{8}\right)$$

$$32 = 144 \sin\left(\frac{\pi}{8}\right)$$

Solving for $\sin\left(\frac{\pi}{8}\right)$, we actually use the initial relationship correctly:

$$h = 12 \sin\left(\frac{\pi}{8}\right)$$

Given $32 = 12h$, we divide both sides by 12 to get:

$$h = \frac{32}{12} = \frac{8}{3}$$

This height corresponds to the perpendicular distance from one base to the opposite side, effectively the 'other two sides' when referring to the height affecting the area directly. However, the question seems to ask for the length of the other pair of sides, not the height, based on the parallelogram's area and an angle. The height calculation helps us understand the parallelogram's dimensions but doesn't directly answer the question about the length of the other two sides.

The confusion comes from interpreting the initial data; the length of the other two sides in a parallelogram, given the area, depends on understanding that the sides opposite each other are equal in length, and the area formula provided involves the height, which was derived indirectly.

However, the lengths of the other two sides are not directly calculated from the area without considering the angle's role in determining the height related to these sides. The formula used to find the height or relate the sides to the given area and angle seems to have been misunderstood.

To clarify, if we're looking to find the lengths of the other two sides based on the provided information (area and one angle), we'd typically use the area formula of a parallelogram in a more directly applicable way if the angle's effect on the side lengths or height were correctly accounted for.

Given the area $A = 32$ and one side of $12$, and wanting the lengths of the sides perpendicular to these, we directly calculate height with the area formula. The actual length of the other two sides, in the context of this explanation, involves understanding that the area and angle given influence the height, which we found as $\frac{8}{3}$, but this height doesn't translate to the length of the side without additional steps that relate the angle to the side lengths, typically through trigonometric relationships in the context of the parallelogram's geometry.

The original attempt to directly solve for the other two sides' lengths mistakenly conflated the height with these side lengths without providing a step that relates the given angle to these lengths. Without additional specific trigonometric relationships or clarifications, the direct calculation of these other side lengths from just the area and one angle provided is not correctly framed.