𝐴𝐵𝐶𝐷 is a parallelogram, where
𝐴𝐵 equals 41 centimeters, 𝐵𝐶 equals 27 centimeters, and the measure of angle 𝐵
is 159 degrees. Find the area of 𝐴𝐵𝐶𝐷, giving
the answer to the nearest square centimeter.
Let’s begin by sketching this
parallelogram. We’re given the measure of one
angle, angle 𝐵, which is 159 degrees. And we’re given the lengths of the
two sides of the parallelogram that enclose this angle. So this part of the parallelogram
will look like this. Of course, the other two sides of
the parallelogram are each parallel to one of the sides we’ve already drawn. And they’re also the same length as
their opposite side. So we can complete the
Now, we’re asked to find the area
of this parallelogram. Usually, we would use the formula
base multiplied by perpendicular height. But we haven’t been given the
perpendicular height of this parallelogram. We could work it out using
trigonometry, but there is another method that we can use. We should recall that the diagonals
of a parallelogram each divide the parallelogram up into two congruent
triangles. If we wish, we can prove this using
the side-side-side or SSS congruency condition.
In triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐶, the
sides 𝐴𝐵 and 𝐶𝐷 are of equal length because they are opposite sides in the
original parallelogram. For the same reason, the sides 𝐴𝐷
and 𝐶𝐵 are also of equal length. 𝐴𝐶 is a shared side in the two
triangles. So we’ve shown that the two
triangles are congruent using the side-side-side condition. As the two triangles are congruent,
their areas are equal. And hence the area of the
parallelogram is twice the area of each triangle.
We then recall the trigonometric
formula for the area of a triangle. In a triangle 𝐴𝐵𝐶, where the
uppercase letters 𝐴, 𝐵, and 𝐶 represent the measures of the three angles in the
triangle and the lowercase letters 𝑎, 𝑏, and 𝑐 represent the lengths of the three
opposite sides, then the trigonometric formula for the area of a triangle is a half
𝑎𝑏 sin 𝐶. Here, 𝑎 and 𝑏 represent the
lengths of any two sides in the triangle and 𝐶 represents the measure of their
included angle. That’s the angle between the two
sides whose length we’re using.
If we consider triangle 𝐴𝐵𝐶 in
our figure then, we know the lengths of the two sides 𝐴𝐵 and 𝐵𝐶. They’re 41 and 27 centimeters,
respectively. And we know the measure of their
included angle; it’s 159 degrees. So substituting 41 and 27 for the
two side lengths in the trigonometric formula and 159 degrees for the measure of
their included angle, we have that the area of triangle 𝐴𝐵𝐶 is a half multiplied
by 41 multiplied by 27 multiplied by sin of 159 degrees.
As we’ve already said, the area of
the parallelogram 𝐴𝐵𝐶𝐷 is twice the area of the individual triangles. So we have two multiplied by a half
multiplied by 41 multiplied by 27 multiplied by sin of 159 degrees. But of course the factor of two and
the factor of a half will cancel each other out, leaving 41 multiplied by 27
multiplied by sin of 159 degrees.
We can now evaluate this on a
calculator, ensuring the calculator is in degree mode. And it gives 396.713
continuing. The question asks us to give our
answer to the nearest square centimeter. So this value rounded to the
nearest integer is 397.
So by recalling that the diagonals
of a parallelogram divide it into two congruent triangles and then applying the
trigonometric formula for the area of a triangle, we found that the area of
parallelogram 𝐴𝐵𝐶𝐷 to the nearest square centimeter is 397 square