WEBVTT
00:00:01.610 --> 00:00:13.250
π΄π΅πΆπ· is a parallelogram, where π΄π΅ equals 41 centimeters, π΅πΆ equals 27 centimeters, and the measure of angle π΅ is 159 degrees.
00:00:13.820 --> 00:00:19.260
Find the area of π΄π΅πΆπ·, giving the answer to the nearest square centimeter.
00:00:20.270 --> 00:00:23.030
Letβs begin by sketching this parallelogram.
00:00:23.360 --> 00:00:28.400
Weβre given the measure of one angle, angle π΅, which is 159 degrees.
00:00:28.660 --> 00:00:34.210
And weβre given the lengths of the two sides of the parallelogram that enclose this angle.
00:00:34.890 --> 00:00:37.970
So this part of the parallelogram will look like this.
00:00:38.450 --> 00:00:44.290
Of course, the other two sides of the parallelogram are each parallel to one of the sides weβve already drawn.
00:00:44.460 --> 00:00:47.360
And theyβre also the same length as their opposite side.
00:00:48.040 --> 00:00:50.060
So we can complete the parallelogram.
00:00:50.590 --> 00:00:53.880
Now, weβre asked to find the area of this parallelogram.
00:00:54.700 --> 00:00:59.100
Usually, we would use the formula base multiplied by perpendicular height.
00:00:59.250 --> 00:01:02.790
But we havenβt been given the perpendicular height of this parallelogram.
00:01:03.220 --> 00:01:08.290
We could work it out using trigonometry, but there is another method that we can use.
00:01:09.020 --> 00:01:16.460
We should recall that the diagonals of a parallelogram each divide the parallelogram up into two congruent triangles.
00:01:17.220 --> 00:01:23.790
If we wish, we can prove this using the side-side-side or SSS congruency condition.
00:01:24.600 --> 00:01:34.400
In triangles π΄π΅πΆ and π΄π·πΆ, the sides π΄π΅ and πΆπ· are of equal length because they are opposite sides in the original parallelogram.
00:01:35.100 --> 00:01:40.890
For the same reason, the sides π΄π· and πΆπ΅ are also of equal length.
00:01:41.880 --> 00:01:44.530
π΄πΆ is a shared side in the two triangles.
00:01:44.690 --> 00:01:49.520
So weβve shown that the two triangles are congruent using the side-side-side condition.
00:01:50.310 --> 00:01:53.560
As the two triangles are congruent, their areas are equal.
00:01:53.670 --> 00:01:58.230
And hence the area of the parallelogram is twice the area of each triangle.
00:01:58.840 --> 00:02:02.180
We then recall the trigonometric formula for the area of a triangle.
00:02:02.630 --> 00:02:21.780
In a triangle π΄π΅πΆ, where the uppercase letters π΄, π΅, and πΆ represent the measures of the three angles in the triangle and the lowercase letters π, π, and π represent the lengths of the three opposite sides, then the trigonometric formula for the area of a triangle is a half ππ sin πΆ.
00:02:22.650 --> 00:02:30.420
Here, π and π represent the lengths of any two sides in the triangle and πΆ represents the measure of their included angle.
00:02:30.610 --> 00:02:34.140
Thatβs the angle between the two sides whose length weβre using.
00:02:35.030 --> 00:02:42.030
If we consider triangle π΄π΅πΆ in our figure then, we know the lengths of the two sides π΄π΅ and π΅πΆ.
00:02:42.150 --> 00:02:45.280
Theyβre 41 and 27 centimeters, respectively.
00:02:45.420 --> 00:02:50.080
And we know the measure of their included angle; itβs 159 degrees.
00:02:50.950 --> 00:03:10.630
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 π΄π΅πΆ is a half multiplied by 41 multiplied by 27 multiplied by sin of 159 degrees.
00:03:11.350 --> 00:03:17.910
As weβve already said, the area of the parallelogram π΄π΅πΆπ· is twice the area of the individual triangles.
00:03:18.150 --> 00:03:26.460
So we have two multiplied by a half multiplied by 41 multiplied by 27 multiplied by sin of 159 degrees.
00:03:27.480 --> 00:03:37.840
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.
00:03:38.300 --> 00:03:43.360
We can now evaluate this on a calculator, ensuring the calculator is in degree mode.
00:03:44.020 --> 00:03:48.750
And it gives 396.713 continuing.
00:03:49.300 --> 00:03:52.910
The question asks us to give our answer to the nearest square centimeter.
00:03:53.580 --> 00:03:58.060
So this value rounded to the nearest integer is 397.
00:03:58.850 --> 00:04:16.580
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 π΄π΅πΆπ· to the nearest square centimeter is 397 square centimeters.