### Video Transcript

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

parallelogram.

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

centimeters.