WEBVTT
00:00:02.310 --> 00:00:16.770
From the figure below, determine the correct inequality from the following: ๐ด๐ต is greater than ๐ถ๐ต, ๐ด๐ต is less than ๐ถ๐ต, ๐ด๐ต is greater than ๐ด๐ถ, or ๐ด๐ถ is less than ๐ถ๐ต.
00:00:19.090 --> 00:00:26.550
Looking at the diagram, we can see that ๐ด๐ต, ๐ถ๐ต, and ๐ด๐ถ all represent the lengths of sides of a triangle.
00:00:27.240 --> 00:00:33.710
Weโve been given four possibilities for relationships that could exist between the lengths of different pairs of sides.
00:00:35.230 --> 00:00:37.570
We havenโt been given any lengths in the diagram.
00:00:37.780 --> 00:00:41.300
Instead, weโve been given some information about some of the angles.
00:00:42.960 --> 00:00:49.250
This suggests that we need to consider the relationship between the lengths of sides and the size of angles in a triangle.
00:00:49.580 --> 00:00:54.870
And therefore, weโre going to approach this question using the angleโside triangle inequality.
00:00:56.710 --> 00:00:59.920
Hereโs what the angleโside triangle inequality tells us.
00:01:00.590 --> 00:01:10.240
If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.
00:01:11.920 --> 00:01:16.690
Basically, what this means is that the longest side of a triangle is opposite the largest angle.
00:01:17.100 --> 00:01:19.590
The shortest side is opposite the smallest angle.
00:01:19.830 --> 00:01:22.470
And the middle side is opposite the middle angle.
00:01:24.180 --> 00:01:28.890
In the diagram, however, weโve only currently got the size of one of the angles in the triangle.
00:01:29.170 --> 00:01:32.260
So we need to consider how we can find the other angles.
00:01:33.680 --> 00:01:36.680
First of all, letโs consider angle ๐ด๐ต๐ถ.
00:01:37.280 --> 00:01:43.950
We can see that the lines ๐ด๐ท and ๐ถ๐ต are parallel as theyโve been marked with blue arrows on their lengths.
00:01:45.290 --> 00:01:49.110
The line ๐ด๐ต is a transversal through these parallel lines.
00:01:49.320 --> 00:01:57.030
And therefore, we can see that the angle ๐ด๐ต๐ถ and the angle of 66 degrees are alternate interior angles.
00:01:57.060 --> 00:01:58.640
Which means that theyโre congruent.
00:02:00.070 --> 00:02:03.900
So angle ๐ด๐ต๐ถ is also 66 degrees.
00:02:05.680 --> 00:02:13.920
Now that we know the measures of two of the angles in the triangle, we can calculate the third because the angle sum in a triangle is always 180 degrees.
00:02:14.440 --> 00:02:22.110
So angle ๐ด๐ถ๐ต can be found by subtracting 52 degrees and 66 degrees from 180 degrees.
00:02:23.670 --> 00:02:25.360
Itโs 62 degrees.
00:02:27.030 --> 00:02:33.610
So now that we know the sizes of all three angles in the triangle, we can deduce something about the lengths of the three sides.
00:02:35.170 --> 00:02:38.140
The largest angle in the triangle is 66 degrees.
00:02:38.310 --> 00:02:44.890
And the angleโside triangle inequality tells us that the longest side of the triangle will be opposite this angle.
00:02:46.280 --> 00:02:49.440
So the longest side of the triangle is the side ๐ด๐ถ.
00:02:51.170 --> 00:02:57.360
The second biggest angle in the triangle is the angle of 62 degrees which is opposite the side ๐ด๐ต.
00:02:58.640 --> 00:03:02.500
This means then that ๐ด๐ต is the second longest side of the triangle.
00:03:04.130 --> 00:03:08.630
The smallest angle of 52 degrees is opposite the shortest side of the triangle.
00:03:08.780 --> 00:03:10.920
So ๐ถ๐ต is the shortest side.
00:03:12.750 --> 00:03:21.290
Now that we have the three sides of the triangle ordered from longest to shortest, we can turn our attention to the four inequalities and determining which are true.
00:03:22.910 --> 00:03:25.440
Firstly, is ๐ด๐ต greater than ๐ถ๐ต?
00:03:25.990 --> 00:03:29.440
Yes, ๐ด๐ต appears above ๐ถ๐ต in the list.
00:03:29.530 --> 00:03:31.810
Which means this first inequality is true.
00:03:33.600 --> 00:03:35.790
Is ๐ด๐ต less than ๐ถ๐ต?
00:03:36.460 --> 00:03:39.900
Well, this is the reverse of the inequality that weโve just shown to be true.
00:03:40.140 --> 00:03:41.900
Therefore, this one must be false.
00:03:43.940 --> 00:03:46.730
Thirdly, is ๐ด๐ต greater than ๐ด๐ถ?
00:03:47.430 --> 00:03:50.480
No, ๐ด๐ถ is the longest side of the triangle.
00:03:50.730 --> 00:03:52.930
So this inequality is also false.
00:03:54.680 --> 00:03:58.030
And finally, is ๐ด๐ถ less than ๐ถ๐ต?
00:03:58.490 --> 00:03:59.940
Again, this is false.
00:04:00.090 --> 00:04:02.550
๐ด๐ถ is the longest side of the triangle.
00:04:04.450 --> 00:04:08.870
So we can conclude that of the four inequalities, only one is true.
00:04:09.270 --> 00:04:11.470
๐ด๐ต is greater than ๐ถ๐ต.