Question Video: Finding the Area of a Parallelogram Using the Trigonometric Formula for Area of Triangles Mathematics

𝐴𝐡𝐢𝐷 is a parallelogram, where 𝐴𝐡 = 41 cm, 𝐡𝐢 = 27 cm, and π‘šβˆ π΅ = 159Β°. Find the area of 𝐴𝐡𝐢𝐷, giving the answer to the nearest square centimeter.

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Video Transcript

𝐴𝐡𝐢𝐷 is a parallelogram, where 𝐴𝐡 equals 41 centimeters, 𝐡𝐢 equals 27 centimeters, and the measure of angle 𝐡 is 159 degrees. Find the area of 𝐴𝐡𝐢𝐷, giving the answer to the nearest square centimeter.

Let’s begin by sketching this parallelogram. We’re given the measure of one angle, angle 𝐡, 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 𝐴𝐡𝐢 and 𝐴𝐷𝐢, the sides 𝐴𝐡 and 𝐢𝐷 are of equal length because they are opposite sides in the original parallelogram. For the same reason, the sides 𝐴𝐷 and 𝐢𝐡 are also of equal length. 𝐴𝐢 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 𝐴𝐡𝐢, where the uppercase letters 𝐴, 𝐡, and 𝐢 represent the measures of the three angles in the triangle and the lowercase letters π‘Ž, 𝑏, and 𝑐 represent the lengths of the three opposite sides, then the trigonometric formula for the area of a triangle is a half π‘Žπ‘ sin 𝐢. Here, π‘Ž and 𝑏 represent the lengths of any two sides in the triangle and 𝐢 represents the measure of their included angle. That’s the angle between the two sides whose length we’re using.

If we consider triangle 𝐴𝐡𝐢 in our figure then, we know the lengths of the two sides 𝐴𝐡 and 𝐡𝐢. 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 𝐴𝐡𝐢 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 𝐴𝐡𝐢𝐷 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 𝐴𝐡𝐢𝐷 to the nearest square centimeter is 397 square centimeters.

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