WEBVTT
00:00:20.500 --> 00:00:25.488
Let's review some information about triangles from geometry.
00:00:25.490 --> 00:00:30.237
Recall that the sum of the angle measures in any triangle is 180 degrees.
00:00:30.239 --> 00:00:31.986
You are going to use that quite often.
00:00:31.988 --> 00:00:34.986
The other thing we can do is classify triangles
00:00:34.988 --> 00:00:38.240
based on either their angle measures or the lengths of their sides.
00:00:38.242 --> 00:00:40.487
Let's look at some definitions.
00:00:40.489 --> 00:00:44.484
An acute triangle is a triangle where all the angles are less than 90 degrees.
00:00:44.486 --> 00:00:47.734
Remember that 90 degrees is the same thing as pi over 2 radians.
00:00:47.736 --> 00:00:51.741
A right triangle is a triangle that has exacly one right angle
00:00:51.743 --> 00:00:53.991
or 90 degrees or pi over 2,
00:00:53.993 --> 00:00:57.738
and an obtuse triangle is a triangle that's got one angle
00:00:57.740 --> 00:01:02.735
that's greater than 90 degrees or pi over 2 radians.
00:01:02.737 --> 00:01:05.889
Now when we classify them based on the lengths of the sides
00:01:05.991 --> 00:01:09.994
we have a scalene triangle, that's one where no sides are congruent,
00:01:09.996 --> 00:01:14.230
isosceles triangle, two sides are congruent,
00:01:14.232 --> 00:01:17.491
and equilateral triangle, all three sides are congruent.
00:01:17.493 --> 00:01:20.742
So let's look at an example where we classify triangles
00:01:20.744 --> 00:01:23.741
based on their angle measures and their sides.
00:01:23.743 --> 00:01:27.986
So let's look at this first example. I can see from the tick marks
00:01:27.988 --> 00:01:31.240
that all three sides are different lengths, so I know it's scalene.
00:01:31.242 --> 00:01:35.242
And then I see one angle measure here that's 2 pi over 3.
00:01:35.244 --> 00:01:39.738
Now you have to know that 2 pi over 3 is bigger than pi over 2.
00:01:39.740 --> 00:01:51.241
So this angle measure is obtuse. So this triangle would be obtuse and scalene.
00:01:51.243 --> 00:01:54.985
All right, let's look at the next example. Here we see a right triangle.
00:01:54.987 --> 00:01:59.237
Remember this symbol means a right angle. We see two sides with one tick mark,
00:01:59.237 --> 00:02:02.494
so we know those two sides are equal and this side is different.
00:02:02.496 --> 00:02:06.987
So we've got a right triangle. Two sides are equal, so it's isosceles.
00:02:06.989 --> 00:02:15.485
So this would be a right, isosceles triangle.
00:02:15.487 --> 00:02:16.993
All right, let's look at the last example.
00:02:16.995 --> 00:02:21.987
Here again I see two sides that are congruent, so I know its isosceles.
00:02:21.989 --> 00:02:25.737
This side is different by the two tick marks, and then I see one angle
00:02:25.739 --> 00:02:30.737
measure that's 70 degrees. One thing that you need to remember from geometry
00:02:30.739 --> 00:02:35.236
is that when you have two sides equal, isosceles triangle, you've got the
00:02:35.238 --> 00:02:37.487
base angles that are equal. Those are the angles opposite
00:02:37.489 --> 00:02:41.239
these congruent sides. So these two angles both have to be equal.
00:02:41.241 --> 00:02:47.240
So since this angle is 70, then this angle is 70. And then of course
00:02:47.242 --> 00:02:49.740
I remember that the sum of the angle measures in a triangle is 180.
00:02:49.742 --> 00:02:56.486
So that gives me 140. This top angle is 40, so I can see that all the angle
00:02:56.488 --> 00:03:10.239
measures are less than 90. So this would be an acute, isosceles triangle.
00:03:12.000 --> 00:03:17.234
The Pythagorean Theorem is one of the most widely recognized theorems
00:03:17.236 --> 00:03:21.770
in all of mathematics. What it states is that the sum of the squares
00:03:21.772 --> 00:03:28.491
of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.
00:03:28.493 --> 00:03:36.027
So here I've got a right triangle drawn. We are going to let the legs be represented with a and b.
00:03:36.029 --> 00:03:39.521
c is the side opposite the right angle. It's the hypotenuse.
00:03:39.523 --> 00:03:43.272
So this is a leg. This is a leg.
00:03:43.274 --> 00:03:49.523
And this is the hypotenuse. So what the Pythagorean Theorem says
00:03:49.525 --> 00:03:54.271
is the sum of the squares of the legs is equal to the square of the hypotenuse.
00:03:54.273 --> 00:03:57.769
And we can use the Pythagorean Theorem to find a missing side of a right triangle.
00:03:57.771 --> 00:04:01.024
You're going to do that a lot and use it a lot. Let's look at some examples.
00:04:01.026 --> 00:04:06.523
In this first example, we know the lengths of the legs, and we are looking for the hypotenuse.
00:04:06.525 --> 00:04:14.270
So we can use this Pythagorean Theorem. We know that 7 squared plus 24 squared equals c squared.
00:04:14.272 --> 00:04:25.000
We can use our calculator. I know that 7 squared is 49. 24 squared is 576.
00:04:25.002 --> 00:04:33.519
And let's see, I'm going to write c squared on the left. 49 plus 576
00:04:33.521 --> 00:04:39.770
is 625. Now of course there are two numbers that when you square you get 625, but since
00:04:39.770 --> 00:04:43.519
we're looking for the length of a side, we will always use the positive number.
00:04:43.521 --> 00:04:50.519
The square root of 625 is 25, so the hypotenuse in this first example is 25.
00:04:50.521 --> 00:04:55.019
Let's look at the next example. Again we've got the lengths of the legs,
00:04:55.021 --> 00:04:58.772
and we are looking for the hypotenuse. So we can use the Pythagorean Theorem.
00:04:58.774 --> 00:05:08.519
I know that c squared is equal to 6 squared plus 5 squared. 6 squared is 36. 5 squared is 25.
00:05:08.521 --> 00:05:13.524
Add that together, and you get 61. Remember we are only going to use the positive
00:05:13.526 --> 00:05:19.774
because it's the length of a side, so c in this example is the square root of 61.
00:05:19.776 --> 00:05:23.526
Now you're always going to make sure that you simplify radicals if you can.
00:05:23.528 --> 00:05:27.522
In this example, 61, you can't simplify, so you'll leave it just like this.
00:05:27.524 --> 00:05:32.269
Let's look at the last example. Here I'm looking for a leg.
00:05:32.271 --> 00:05:36.769
I know one leg is the square root of 2, and I know the hypotenuse is the square root of 11.
00:05:36.771 --> 00:05:42.272
So using the Pythagorean theorem we have b squared plus the square root of 2 squared
00:05:42.274 --> 00:05:48.270
equals the square root of 11 squared. OK, so that gives us b squared,
00:05:48.272 --> 00:05:51.520
remember when you square a square root you get the number under the radical,
00:05:51.522 --> 00:05:56.769
so I have 2 and then the square root of 11 squared is 11.
00:05:56.771 --> 00:06:04.271
Now we'll subtract 2 from both sides and get 9, so b in this case is 3.
00:06:04.271 --> 00:06:09.277
So you're going to use the Pythagorean theorem many, many times to find a missing side of a right triangle.
00:06:09.279 --> 00:06:15.000
It's important that you know the Pythagorean theorem, and remember it only applies to a right triangle.
00:06:16.515 --> 00:06:20.568
Let's take a look at similar triangles.
00:06:20.570 --> 00:06:25.567
Similar triangles are triangles that have the same shape, but not necessarily the same size.
00:06:25.569 --> 00:06:31.320
There are two properties of similar triangles. Number one, the corresponding angles have the same measure,
00:06:31.322 --> 00:06:36.318
and the second, the ratios of the corresponding sides are equal.
00:06:36.320 --> 00:06:39.570
In other words, the corresponding sides are proportional.
00:06:39.572 --> 00:06:42.819
Let's look at this example. I've got two similar triangles.
00:06:42.821 --> 00:06:48.316
I can see that I have angle measures that have the same measure by the tick marks.
00:06:48.318 --> 00:06:50.070
A has the same measure as D,
00:06:50.072 --> 00:06:52.069
B has the same measure as E,
00:06:52.071 --> 00:06:54.820
and C has the same measure as F.
00:06:54.820 --> 00:06:58.818
We also know that the ratios of the corresponding sides are equal.
00:06:58.820 --> 00:7:02.818
So corresponding sides are the sides that are opposite the congruent angles,
00:07:02.820 --> 00:07:04.819
the angles that have the same measure.
00:07:04.821 --> 00:07:10.060
So a corresponds to d so that ratio is the same as the
00:07:10.065 --> 00:07:12.571
ratio of b which corresponds to e
00:07:12.573 --> 00:07:17.060
which is the same as the ratio of c which corresponds to f.
00:07:17.063 --> 00:07:20.318
All of these ratios will be equal in similar triangles.
00:07:21.320 --> 00:07:26.560
Now we talked about being proportional. Let's think about this.
00:07:26.565 --> 00:07:28.569
We have something called the proportionality constant.
00:07:28.571 --> 00:07:30.070
We use the letter k,
00:07:30.072 --> 00:07:33.570
where k is greater than or equal to 1.
00:07:33.572 --> 00:07:37.567
So the proportionality constant is equal to the length
00:07:37.569 --> 00:07:41.317
of a side of the larger triangle divided by the length
00:07:41.319 --> 00:07:45.069
of the corresponding side of the smaller triangle.
00:07:45.071 --> 00:07:48.570
If the triangles are the same size, then k is 1.
00:07:48.572 --> 00:07:53.321
So let's look at this example and find the proportionality constant k.
00:07:53.323 --> 00:07:58.818
Well we can see that we've got angle measures that are the same, again by the tick marks.
00:07:58.820 --> 00:08:01.818
A and D have the same measure, B and E have the same measure,
00:08:01.820 --> 00:08:04.069
C and F have the same measure.
00:08:04.071 --> 00:08:07.317
So these sides correspond because they're opposite
00:08:07.319 --> 00:08:10.572
congruent angles that have the same measure.
00:08:10.574 --> 00:08:14.820
So in order to find that proportionality constant we want to know
00:08:14.822 --> 00:08:18.820
the larger side which is 8
00:08:18.822 --> 00:08:22.572
divided by the corresponding side in the smaller triangle, 4.
00:08:22.574 --> 00:08:24.815
Well that happens to be 2.
00:08:24.817 --> 00:08:27.818
So in this example our proportionality constant is 2.
00:08:27.820 --> 00:08:32.819
And we can use proportionality constants to find missing sides of similar triangles.
00:08:32.821 --> 00:08:35.319
Let's look at some examples of that.
00:08:37.819 --> 00:08:41.820
Let's look at some examples of finding missing sides
00:08:41.822 --> 00:08:46.322
of similar right triangles by finding the proportionality constant, k.
00:08:46.324 --> 00:08:49.810
In this first right triangle we see we have two legs.
00:08:49.812 --> 00:08:51.000
We know the lengths of the legs.
00:08:51.002 --> 00:08:54.319
We can use the Pythagorean theorem to find the hypotenuse.
00:08:54.321 --> 00:09:02.839
So let's do that. So AB squared is equal to 56 squared plus 33 squared.
00:09:02.839 --> 00:09:12.339
So 56 squared is 3136,
00:09:12.341 --> 00:09:21.591
and 33 squared is 1089. Add those together.
00:09:21.593 --> 00:09:29.586
We get AB squared is 4225. We'll take the square root of that,
00:09:29.588 --> 00:09:38.570
and we get AB is 65. So now we've found the missing side of the first triangle.
00:09:38.572 --> 00:09:41.817
Now we know the triangles are similar.
00:09:41.819 --> 00:09:45.817
So this is 65.
00:09:45.819 --> 00:09:51.583
We can find that proportionality constant by taking the only side I know in the second triangle,
00:09:51.585 --> 00:09:58.337
the larger triangle, 260. Well this side, this hypotenuse, corresponds to 65,
00:09:58.339 --> 00:10:07.337
the hypotenuse in the first triangle, and 260 divided by 65 is 4.
00:10:07.339 --> 00:10:13.085
So my proportionality constant is 4. So that's going to make finding XZ and YZ easy.
00:10:13.087 --> 00:10:22.838
All I have to do to find XZ is multiply its corresponding side in the smaller triangle, 56, by 4.
00:10:22.840 --> 00:10:27.836
56 times 4, let's just do that on our calculator.
00:10:27.838 --> 00:10:41.333
56 times 4 is 224. So that's the length of XZ, and then to find YZ you're going to find
00:10:41.335 --> 00:10:49.336
its corresponding side is 33. So we're going to multiply 33 by 4,
00:10:49.338 --> 00:10:55.836
and we get 132. So those are the missing sides of that first example.
00:10:55.838 --> 00:10:58.086
Let's look at another example.
00:10:58.334 --> 00:11:01.340
In the second example, the triangles are similar again,
00:11:01.342 --> 00:11:04.585
but notice that they're not oriented in the same direction.
00:11:04.587 --> 00:11:09.584
So what you might want to do to help you is let's redraw this triangle here.
00:11:09.586 --> 00:11:21.000
We can draw it. Let's see. This is R...
00:11:21.010 --> 00:11:23.835
So now the triangles are oriented in the same direction.
00:11:23.837 --> 00:11:26.084
Let's see which side we knew. RS is 1.
00:11:26.086 --> 00:11:29.335
So now, again we have similar triangles.
00:11:29.337 --> 00:11:34.590
In the second right triangle, we know a leg and the hypotenuse,
00:11:34.592 --> 00:11:37.834
so we can use the Pythagorean theorem to find this.
00:11:37.836 --> 00:11:46.332
Let's call this KWP, and so let's see.
00:11:46.334 --> 00:11:51.085
We have KP squared plus the other leg [squared] is 8 [squared]
00:11:51.087 --> 00:11:54.334
is equal to the hypotenuse squared. Remember the Pythagorean theorem.
00:11:54.336 --> 00:12:03.585
So working and solving for KP, 17 squared minus 8 squared is 225.
00:12:03.587 --> 00:12:10.835
And the square root of that I believe is 15, but let me check. Yes.
00:12:10.837 --> 00:12:16.086
So the missing side of the second triangle, this side, is 15.
00:12:16.088 --> 00:12:23.586
So now we need to find in the smaller triangle we need to find RT and ST.
00:12:23.588 --> 00:12:27.841
That's going to be easy because we know a side in the smaller triangle is 1.
00:12:27.843 --> 00:12:33.335
1 corresponds to the 8, so the proportionality constant in this example
00:12:33.337 --> 00:12:36.585
would be 8 over 1 which of course is 8.
00:12:36.587 --> 00:12:42.585
So in order to find the sides in the smaller triangle we take the side of the larger triangle, divide by 8.
00:12:42.587 --> 00:12:48.334
So RT corresponds to 15, so I have 15 divided by 8.
00:12:48.336 --> 00:12:52.300
And ST, the hypotenuse here, corresponds to this hypotenuse,
00:12:52.332 --> 00:12:59.085
so I take 17 divided by 8 to find those missing sides in the smaller triangle.
00:13:01.086 --> 00:13:06.336
There are two special right triangles whose properties are important for you to know.
00:13:06.338 --> 00:13:12.084
Let's look at the first one. The first one is pi over 4, pi over 4, pi over 2
00:13:12.086 --> 00:13:16.385
or a 45, 45, 90 right triangle. I've got one drawn here.
00:13:16.387 --> 00:13:20.635
We can see that the legs are equal. Remember, if the legs are equal we know
00:13:20.637 --> 00:13:24.635
that this angle is pi over 4 and this angle is pi over 4,
00:13:24.637 --> 00:13:29.136
so we've got an isosceles right triangle. We know the legs are the same length,
00:13:29.138 --> 00:13:31.139
so I've labelled them with the letter a.
00:13:31.141 --> 00:13:36.137
Now I've labelled the hypotenuse c. So we can use the Pythagorean theorem,
00:13:36.139 --> 00:13:41.135
and we can find the length of c. So we know that c squared, the hypotenuse squared,
00:13:41.137 --> 00:13:51.387
is leg squared plus leg squared. Well a squared plus a squared is 2 a squared.
00:13:51.389 --> 00:13:55.389
Now to find c remember we're only choosing the positive. We're going to take the
00:13:55.391 --> 00:14:01.637
square root of 2 a squared. a and c are both positive. So this is going
00:14:01.639 --> 00:14:07.386
to be the square root of 2 times a.
00:14:07.386 --> 00:14:13.136
So in any pi over 4, pi over 4, pi over 2, or 45, 45, 90 triangle,
00:14:13.138 --> 00:14:25.634
the hypotenuse is always the square root of 2 times the length of the leg.
00:14:25.636 --> 00:14:28.638
That's an important thing for you to know.
00:14:28.640 --> 00:14:32.637
If you need to find the leg you would take the hypotenuse and divide by the square root of 2.
00:14:32.639 --> 00:14:35.887
But this relationship is very important for you to know.
00:14:35.889 --> 00:14:46.136
We can let a be any number. So let's draw that same triangle, and we'll let a equal 1.
00:14:46.138 --> 00:14:52.635
So the legs are 1. And then what would the hypotenuse be?
00:14:52.637 --> 00:14:57.634
Well, it's going to be the length of the leg, which is 1, times the square root of 2
00:14:57.636 --> 00:15:00.885
which is the square root of 2. And of course you can use the Pythagorean theorem,
00:15:00.887 --> 00:15:03.638
and this would be the square root of 1 squared plus 1 squared which of course
00:15:03.640 --> 00:15:09.130
is the square root of 2. This is a very important triangle for you to remember.
00:15:09.132 --> 00:15:13.635
You're going to use it throughout the course. So remember this whole triangle,
00:15:13.637 --> 00:15:18.385
the pi over 4, the ones on the legs, and the square root of 2 for the hypotenuse.
00:15:18.389 --> 00:15:21.887
Now let's look at another special right triangle.
00:15:24.139 --> 00:15:28.816
Let's look at a second important special right triangle that you should know.
00:15:28.818 --> 00:15:34.071
It's the pi over 6, pi over 3, pi over 2 or the 30, 60, 90 right triangle.
00:15:34.073 --> 00:15:38.070
To get the sides we're going to start with an equilateral triangle. Remember that
00:15:38.072 --> 00:15:43.069
means all angles are equal and all the sides are equal. So I've got the tick marks
00:15:43.071 --> 00:15:47.569
denoting all the sides are equal. We're going to start by drawing a height
00:15:47.571 --> 00:15:53.820
from one of the vertices. Now when it's an equilateral triangle, that height bisects
00:15:53.822 --> 00:15:59.569
both the angle up here and the side down here. So that pi over 3 angle got bisected
00:15:59.571 --> 00:16:05.569
into pi over 6 and pi over 6. Of course this is a right angle because it's a height.
00:16:05.571 --> 00:16:12.066
Now if we let this side be 2a, then remember we've bisected this side, so
00:16:12.068 --> 00:16:18.067
this little piece is a. So now we have that pi over 6, pi over 3, pi over 2 triangle.
00:16:18.069 --> 00:16:26.067
Let's just draw that piece, that triangle. Pi over 6, pi over 3,
00:16:26.069 --> 00:16:31.071
this is 2a, this is a. And let's call this h, this is the height.
00:16:31.073 --> 00:16:35.821
So we can find that third side in the right triangle using the Pythagorean theorem.
00:16:35.823 --> 00:16:43.320
We have h squared plus a squared equals hypotenuse squared, 2a squared.
00:16:43.322 --> 00:16:51.320
Well let's see, when you square 2a what do you get? You get 4 a squared.
00:16:51.322 --> 00:16:56.818
Now we'll subtract a squared from both sides. We get 3 a squared.
00:16:56.820 --> 00:17:02.819
So h is going to be the square root of 3 a squared. Remember a is positive.
00:17:02.821 --> 00:17:07.319
So this is going to be the square root of 3 times a.
00:17:07.321 --> 00:17:12.570
So the side opposite the pi over 3 is going to be the square root of 3
00:17:12.572 --> 00:17:19.315
times the side opposite the pi over 6. So this is an important relationship as well.
00:17:19.317 --> 00:17:30.568
So what you see in this triangle, the hypotenuse is 2 times the leg opposite
00:17:30.570 --> 00:17:41.570
the pi over 6 angle. So the hypotenuse is always 2 times this side.
00:17:41.572 --> 00:17:54.071
And the leg opposite the pi over 3, or 60 degree angle, is always the square root of 3
00:17:54.073 --> 00:18:02.819
times the leg opposite the pi over 6 angle. So again these are important relationships
00:18:02.821 --> 00:18:09.321
for you to remember in this right triangle. Hypotenuse is always 2 times
00:18:09.323 --> 00:18:12.570
the leg opposite the pi over 6, and the leg opposite the pi over 3
00:18:12.572 --> 00:18:15.318
is always the square root of 3 times this side.
00:18:15.320 --> 00:18:21.569
Now we can let a be any number. So what we're going to do is, what if we let a equal 1?
00:18:21.571 --> 00:18:27.822
Then what are we going to get in that triangle? Well, here's my pi over 3.
00:18:27.824 --> 00:18:32.320
Here's my pi over 6. So we're letting a be 1. So this side will be 1.
00:18:32.322 --> 00:18:37.569
Hypotenuse will be 2, and this side will be the square root of 3 times 1 which of course
00:18:37.571 --> 00:18:43.560
is the square root of 3. This is another, or your second, important special right triangle
00:18:43.562 --> 00:18:47.071
that you should remember. It's important that you know you've got to put your angles
00:18:47.073 --> 00:18:50.822
in your triangle. You have to remember to put the 1 opposite the pi over 6,
00:18:50.824 --> 00:18:55.068
the square root of 3 opposite the pi over 3, and the 2 on the hypotenuse.
00:18:55.070 --> 00:18:57.821
We're going to use these a lot throughout this course.
00:18:57.823 --> 00:19:03.070
Now let's look at some examples where we use these special right triangles.
00:19:05.317 --> 00:19:10.468
Let's look at some examples where we find the lengths of missing sides using
00:19:10.470 --> 00:19:13.718
techniques we learned earlier about similar triangles.
00:19:14.000 --> 00:19:18.469
Let's look at this first right triangle. We see we've got a right angle. We see
00:19:18.471 --> 00:19:25.450
one angle is marked as pi over 4, so the third angle in that triangle has to be pi over 4.
00:19:25.452 --> 00:19:31.239
So this is one of our special triangles, our pi over 4, pi over 4, pi over 2 triangle.
00:19:31.241 --> 00:19:36.218
We want to find the lengths of the missing sides. Well, you should remember
00:19:36.220 --> 00:19:45.469
one of those triangles, the pi over 4, pi over 4 triangle. We can let the lengths
00:19:45.471 --> 00:19:49.469
of the legs be 1, and remember the hypotenuse is the square root of 2.
00:19:49.471 --> 00:19:53.968
So this is that special triangle you should have memorized. And now we can use that
00:19:53.970 --> 00:19:57.969
proportionality constant we talked about earlier. We want to find the length of BC.
00:19:57.971 --> 00:20:03.968
That's pretty easy because we know the legs are equal, so I know BC has to be 5.
00:20:04.000 --> 00:20:11.969
Now we want to find the hypotenuse. Well, the proportionality constant is 5
00:20:11.971 --> 00:20:18.960
because 5 and 1 correspond. So to find AB we're going to multiply
00:20:18.964 --> 00:20:22.218
the proportionality constant, 5, times the square root of 2,
00:20:22.220 --> 00:20:29.468
so AB will be 5 square roots of 2. Let's look at another example.
00:20:29.470 --> 00:20:35.218
So here we see a right triangle again. We've got one angle marked as pi over 6, so
00:20:35.220 --> 00:20:39.468
we know the third angle in that right triangle has got to be pi over 3.
00:20:39.470 --> 00:20:44.468
So this is our special triangle, pi over 6, pi over 3, pi over 2 triangle.
00:20:44.470 --> 00:20:51.860
Well, you should remember your special triangle where this is pi over 6, the side opposite
00:20:51.862 --> 00:20:57.468
is 1, the hypotenuse is 2, and the side opposite the pi over 3 is the square root of 3.
00:20:57.470 --> 00:21:04.234
So that's a triangle you should have memorized. And now we see that 11 corresponds to 2,
00:21:04.236 --> 00:21:13.000
so here I can find the proportionality constant is 11/2. So in order to find BC
00:21:13.002 --> 00:21:17.214
we're going to multiply 11/2 times the corresponding side in this triangle.
00:21:17.216 --> 00:21:23.720
So BC will be 1 times 11/2 which of course is just 11/2.
00:21:23.722 --> 00:21:29.236
Now we'll find the last side, AC. Well AC corresponds to the square root of 3,
00:21:29.238 --> 00:21:33.219
so we're going to take 11/2 again, the proportionality constant,
00:21:33.221 --> 00:21:38.468
times the square root of 3. So it's going to be 11 square roots of 3 over 2
00:21:38.470 --> 00:21:42.740
for the length of AC. Let's look at one more.
00:21:43.719 --> 00:21:48.968
This last example again we see one angle is a right angle. We see one angle is pi over 3.
00:21:48.970 --> 00:21:53.231
So we know the third angle in this triangle is pi over 6,
00:21:53.233 --> 00:21:59.468
and we know the leg opposite the pi over 3 is the square root of 15. Again,
00:21:59.470 --> 00:22:04.725
we can just use this triangle here. So the sides that correspond are the square root of 3
00:22:04.727 --> 00:22:09.216
and the square root of 15 because they're opposite that pi over 3. So in this example,
00:22:09.218 --> 00:22:14.460
the proportionality constant is the square root of 15 divided by the square root of 3
00:22:14.462 --> 00:22:19.469
which is the square root of 5. So in order to find the other missing sides,
00:22:19.471 --> 00:22:22.469
we're going to multiply the corresponding side in the smaller triangle
00:22:22.471 --> 00:22:30.738
by the square root of 5. So BC will be 1 times the square root of 5 which of course
00:22:30.740 --> 00:22:39.218
is just the square root of 5, and then AB, the hypotenuse, is going to be
00:22:39.220 --> 00:22:46.960
2 times the square root of 5. So we've found all the missing sides
00:22:46.962 --> 00:22:50.730
of these special right triangles. This is very important because you will use
00:22:50.732 --> 00:22:55.467
throughout this course these triangles and find missing sides.
00:22:55.469 --> 00:22:58.968
So it's important that you remember those relationships, you remember these triangles,
00:22:58.970 --> 00:23:03.222
and you can find missing sides pretty easily.
00:23:03.224 --> 00:23:09.976
You should practice all the examples that we've done today so that you can do them on your own.