WEBVTT mathematics/geometry/pyo
00:00:00.000 --> 00:00:02.800
Welcome back to Educator.com.
00:00:02.800 --> 00:00:11.200
In this next lesson, we are going to talk about triangles and some different ways we can classify them.
00:00:11.200 --> 00:00:27.100
A triangle is a three-sided polygon; now, a polygon is any figure that has sides and is closed.
00:00:27.100 --> 00:00:36.400
A polygon can look like that, as long as it is closed, and all of the sides are straight.
00:00:36.400 --> 00:00:47.500
If I have that, it would not be considered a polygon; if I have just maybe something like this,
00:00:47.500 --> 00:00:55.700
that is not a polygon, either, because it is not closed--that is not an example of a polygon.
00:00:55.700 --> 00:01:06.000
A triangle is a polygon with three sides; now, the triangle is made up of sides, vertices, and angles.
00:01:06.000 --> 00:01:22.900
The sides are these right here, side AB, side BC, and side CA (or AC).
00:01:22.900 --> 00:01:38.400
The vertices are the points; the word "vertices" is the plural of "vertex," so if I just talk about one, then I would just say "vertex."
00:01:38.400 --> 00:01:56.600
But since I have three, it is "vertices"; and that would just be point A, point B, and point C--those are my vertices.
00:01:56.600 --> 00:02:07.600
The angles: now, for this angle right here, let's say, I can say "angle ABC," or I can just say "angle B,"
00:02:07.600 --> 00:02:13.900
because as long as there is only one angle...if I have a line that is coming out through here,
00:02:13.900 --> 00:02:17.500
then I will have several different angles, so I can't label it angle B;
00:02:17.500 --> 00:02:27.600
but as long as there is only one single angle from that vertex, then you can label the angle by that point.
00:02:27.600 --> 00:02:36.700
This angle right here can be called angle B; this angle can be called angle A, since there is no other angle there.
00:02:36.700 --> 00:02:50.100
This angle can be called angle C, or you can just do it the other way: angle ABC, angle BCA, angle BAC.
00:02:50.100 --> 00:03:01.400
But this is the easiest way: angle B, angle A, and angle C; that is the triangle.
00:03:01.400 --> 00:03:11.100
Now, to classify triangles, we can classify them in two ways: by their angles and by their sides.
00:03:11.100 --> 00:03:20.500
These are the ways that we can classify the triangles by their angles, meaning that, based on the angles of the triangle, we have different names for them.
00:03:20.500 --> 00:03:29.600
The first one: an **acute triangle**...well, we know that an acute angle is an angle that measures less than 90, smaller than 90.
00:03:29.600 --> 00:03:45.900
So, an acute triangle is a triangle where all of the angles are acute; all of the angles measure less than 90.
00:03:45.900 --> 00:03:51.200
The next one: **obtuse triangle**: one angle is obtuse.
00:03:51.200 --> 00:04:02.700
Now, all triangles have at least two acute angles.
00:04:02.700 --> 00:04:13.600
The way we classify these other ones: for this one, an obtuse angle is an angle that measures greater than 90;
00:04:13.600 --> 00:04:19.700
only one of them will be obtuse--there is no way that you can have a triangle with two obtuse angles.
00:04:19.700 --> 00:04:28.400
That means two of the angles (since we have three) are going to be acute, and then one of them is going to be obtuse.
00:04:28.400 --> 00:04:45.500
But all triangles must have at least two acute angles; so here is an obtuse triangle; here is your obtuse angle, and then your acute angles.
00:04:45.500 --> 00:05:00.500
The same thing for the next one, a **right triangle**: a right triangle is when only one angle is a right angle, like this.
00:05:00.500 --> 00:05:03.700
Again, the other two angles must be acute.
00:05:03.700 --> 00:05:22.600
Now, if I try to draw two angles of a triangle to be right, see how there is no way that that could be a triangle,
00:05:22.600 --> 00:05:39.900
because a triangle, remember, has to have three sides; "tri" in triangle means three.
00:05:39.900 --> 00:05:52.300
Now, an **equiangular triangle** means that all of the angles are equal--"equal angle"--can you see the two words formed in there?
00:05:52.300 --> 00:05:57.100
So, equiangular is when all of the angles are congruent.
00:05:57.100 --> 00:06:08.300
Now, if all of the angles are congruent, then each angle is going to measure 60 degrees.
00:06:08.300 --> 00:06:18.500
Now, we are going to go over this later on; but the three angles of a triangle have to add up to 180.
00:06:18.500 --> 00:06:30.300
If all three angles are congruent, then I just do 180 divided by 3, because each angle has to have the same number of degrees.
00:06:30.300 --> 00:06:36.300
So, 180 divided by 3 is going to be 60; so this will be 60, 60, and then 60.
00:06:36.300 --> 00:06:52.100
And that is an equiangular triangle; and of course, an equiangular triangle is an acute triangle, because 60 degrees is an acute angle; all three angles are acute.
00:06:52.100 --> 00:06:58.800
In an equiangular triangle, all angles are congruent; they all measure 60 degrees.
00:06:58.800 --> 00:07:05.800
OK, so then, classifying triangles by size: we just went over by angles, depending on how the angles look--
00:07:05.800 --> 00:07:13.100
if it is a right angle in the triangle, an obtuse angle, or an acute angle.
00:07:13.100 --> 00:07:18.300
We can also classify triangles by the size.
00:07:18.300 --> 00:07:36.300
If I have a triangle where no two sides are congruent--all three sides are different lengths (like this is 3, 4, 5), then this is a **scalene triangle**.
00:07:36.300 --> 00:07:45.100
So, I can show this by making little slash marks; I can make this once; then this, I will do twice to show that these are not congruent;
00:07:45.100 --> 00:07:53.500
and then I will do this one three times--that shows that none of these sides are congruent to each other.
00:07:53.500 --> 00:07:57.100
And that is a scalene triangle.
00:07:57.100 --> 00:08:13.200
The next one, an **isosceles triangle**, is when I have at least two sides being congruent.
00:08:13.200 --> 00:08:19.200
And then, an **equilateral triangle** is when all of the sides are congruent.
00:08:19.200 --> 00:08:24.700
So, an equilateral triangle will be considered an isosceles triangle, because any time you have at least
00:08:24.700 --> 00:08:29.700
(meaning two or three) sides being congruent, then it is considered isosceles.
00:08:29.700 --> 00:08:33.900
But this is the more specific name if all three are congruent.
00:08:33.900 --> 00:08:43.500
So, you would call this an equilateral triangle; but it is also considered an isosceles triangle.
00:08:43.500 --> 00:08:52.500
Now, remember this one: equilateral; equiangular is when all of the angles are congruent, and equilateral is when all of the sides are congruent.
00:08:52.500 --> 00:09:00.300
These are some ways you can classify triangles by the sides.
00:09:00.300 --> 00:09:09.300
Isosceles triangle: I see here that these two sides are congruent, which is an isosceles triangle.
00:09:09.300 --> 00:09:19.900
If I label this as triangle ABC, then I can say that triangle ABC is isosceles.
00:09:19.900 --> 00:09:25.900
Any time you have a triangle, you can label it like this, triangle ABC, just like when you have an angle, you can say angle A.
00:09:25.900 --> 00:09:33.000
So, if you have a triangle, you are going to say triangle ABC.
00:09:33.000 --> 00:09:47.500
This right here, this side that is not congruent to the other two sides, is called your base.
00:09:47.500 --> 00:09:58.200
These two sides that are congruent are called your legs.
00:09:58.200 --> 00:10:09.600
Now, the two angles that are formed from the legs to the base are called base angles.
00:10:09.600 --> 00:10:17.800
Base angles would be this angle right here and this angle right here.
00:10:17.800 --> 00:10:27.400
And then, this angle right here, or that point, is called your vertex.
00:10:27.400 --> 00:10:34.200
This right here, that is formed from the two congruent legs, is called your vertex.
00:10:34.200 --> 00:10:41.300
So, we have legs; we have a vertex; we have the base; and then, you have base angles.
00:10:41.300 --> 00:10:44.100
And this makes up your isosceles triangle.
00:10:44.100 --> 00:11:07.700
And then, just to go over the right triangle: these are also called your legs, but this one is not called the base; it is called a hypotenuse.
00:11:07.700 --> 00:11:12.300
That is just to review over that.
00:11:12.300 --> 00:11:19.200
In the isosceles triangle, these two are congruent, so they are called your legs; this one is your base.
00:11:19.200 --> 00:11:24.500
Now, don't think that the base is the side always on the bottom; no.
00:11:24.500 --> 00:11:34.000
It depends on the triangle: I can move this triangle around and make it look like this, and I can say that these two sides are congruent.
00:11:34.000 --> 00:11:41.300
Then, this would be my base, and these are my legs.
00:11:41.300 --> 00:11:53.900
Find x, AB, BC, and AC; let me just write this to show that they are my segments.
00:11:53.900 --> 00:12:02.100
To solve this, if I want to solve for x, I know that these two are congruent.
00:12:02.100 --> 00:12:05.900
Since they are congruent, I can just make these equal to each other.
00:12:05.900 --> 00:12:22.200
2x + 3 = 3x - 2; then, if I subtract the 3x, I get -x =...I subtract 3 over, so I get -5; x is 5.
00:12:22.200 --> 00:12:32.700
There is my x; AB is (and I am not going to put a line over this one, because I am finding the lengths;
00:12:32.700 --> 00:12:51.800
if I am finding the measure, then I don't put the line over it) 2 times 5, plus 3; so that is 10 + 3, is 13.
00:12:51.800 --> 00:13:03.100
And then, for BC, since I know x, even though this side is the base, and it doesn't have anything to do with these sides;
00:13:03.100 --> 00:13:14.200
since I know x, I can solve for BC; that is 5 + 1, so BC = 6.
00:13:14.200 --> 00:13:31.500
And then, AC is 3(5) - 2, so that is 15, minus 2 is 13; and that is everything.
00:13:31.500 --> 00:13:37.800
So again, if you have an isosceles triangle, and they want you to solve for x, then you can just say that,
00:13:37.800 --> 00:13:46.500
since these two sides are congruent, you can make these two congruent.
00:13:46.500 --> 00:13:54.900
All right, let's go over a few examples now: Classify each triangle with the given angle measures by its angles and sides.
00:13:54.900 --> 00:14:04.700
We have to classify each of these triangles with the given measures in two ways: by its angles and by its sides.
00:14:04.700 --> 00:14:19.600
The first one: the angles are here, and the sides here.
00:14:19.600 --> 00:14:35.000
To classify this triangle by its angles, look: I have one that is really big--that is an obtuse angle; that means that this would be an obtuse triangle.
00:14:35.000 --> 00:14:37.500
I am just going to draw a little triangle right there.
00:14:37.500 --> 00:14:46.500
And then, by its sides, remember: sides are scalene, isosceles, and equilateral.
00:14:46.500 --> 00:14:56.700
No two sides are congruent (the same), so this would be a scalene triangle.
00:14:56.700 --> 00:15:05.100
The next one: 50, 50, and 80: by its angles--look at the angle measures: they are all acute angles.
00:15:05.100 --> 00:15:11.900
So then, this triangle would be an acute triangle.
00:15:11.900 --> 00:15:15.500
And then, with its sides, are any of them the same?
00:15:15.500 --> 00:15:25.800
We have two that are the same; these two are the same; then, I have an isosceles triangle.
00:15:25.800 --> 00:15:46.200
The next one: this is an acute triangle, because they are all the same; but more specifically, this would be an equiangular triangle.
00:15:46.200 --> 00:16:04.300
And then, since they are all the same, even though it could be isosceles (because isosceles is two or more), a more specific name would be equilateral.
00:16:04.300 --> 00:16:15.100
And the last one: 30, 60, 90: well, I have one angle that is a right angle, so this is a right triangle.
00:16:15.100 --> 00:16:29.300
And then, my sides: this would be scalene, because they are all different.
00:16:29.300 --> 00:16:35.900
The next example: We are going to fill in the blanks with "always," "sometimes," or "never."
00:16:35.900 --> 00:16:41.300
Scalene triangles are [always/sometimes/never] isosceles.
00:16:41.300 --> 00:16:45.800
Well, we know that a scalene triangle is when they are all different.
00:16:45.800 --> 00:16:50.800
An isosceles triangle is when we have two or more sides that are the same.
00:16:50.800 --> 00:17:02.100
So, a scalene triangle would never be isosceles, because in order for the triangle to be scalene, all of the sides have to be different.
00:17:02.100 --> 00:17:13.700
In order for the triangle to be isosceles, you have to have two or more the same, so this would be "never" isosceles.
00:17:13.700 --> 00:17:21.900
The next one: Obtuse triangles [always/sometimes/never] have two obtuse angles.
00:17:21.900 --> 00:17:30.300
Well, if I draw an obtuse angle here, and then I draw an obtuse angle here, this has to be greater than 90,
00:17:30.300 --> 00:17:38.400
and this has to be greater than 90; a triangle has to have three sides only, so there is no way for that to be a triangle.
00:17:38.400 --> 00:17:46.300
So, this is "never."
00:17:46.300 --> 00:17:54.700
Equilateral triangles are [always/sometimes/never] acute triangles.
00:17:54.700 --> 00:18:01.800
If they are equilateral, that means that it has to be like this.
00:18:01.800 --> 00:18:11.800
Can we ever have an obtuse triangle where they are the same?--no, because you know that this can only be extra long.
00:18:11.800 --> 00:18:19.300
And then, if we have a right triangle, no, because we can't have the hypotenuse be the same length as one of the legs.
00:18:19.300 --> 00:18:29.700
So, equilateral triangles are "always" acute.
00:18:29.700 --> 00:18:35.300
Isosceles triangles are [always/sometimes/never] equilateral triangles.
00:18:35.300 --> 00:18:47.100
An isosceles triangle is like this or like this; "isosceles" can be two or three sides being congruent.
00:18:47.100 --> 00:18:51.100
Now, sometimes they are like this, and sometimes they are like that.
00:18:51.100 --> 00:18:57.400
That means that sometimes they are going to be equilateral.
00:18:57.400 --> 00:19:11.500
Because isosceles triangles can be considered equilateral triangles (but not always; if it is only two, then it is not; if it is three, then it is), it is "sometimes."
00:19:11.500 --> 00:19:18.500
Acute triangles are [always/sometimes/never] equiangular.
00:19:18.500 --> 00:19:30.600
If I have an acute triangle, yes, it could be equiangular; but can I draw an acute triangle that is not equiangular?
00:19:30.600 --> 00:19:44.100
How about like this? This is not equiangular, but they are all acute angles.
00:19:44.100 --> 00:20:00.100
Sometimes it could be like this, or sometimes it could be like this; "sometimes"--acute triangles are sometimes equiangular.
00:20:00.100 --> 00:20:14.800
If I have, let's say, 50, 50, and 80; see how these are all acute angles.
00:20:14.800 --> 00:20:23.600
They are acute angles, but then it is not equiangular; only if they are 60, 60, 60 are they equiangular.
00:20:23.600 --> 00:20:30.800
I can have other acute triangles that are not equiangular.
00:20:30.800 --> 00:20:37.800
OK, the next example: Find x and all of the sides of the isosceles triangle.
00:20:37.800 --> 00:20:43.700
Here, it is this side and this side that are congruent.
00:20:43.700 --> 00:20:50.300
Now again, this is the base (even though it is not at the bottom, it is still called the base).
00:20:50.300 --> 00:20:56.200
These are my legs, the vertex, and the base angles.
00:20:56.200 --> 00:21:07.000
I am going to make 9x + 12 equal to 11x - 4, because those sides are congruent.
00:21:07.000 --> 00:21:21.300
Then I am going to solve this out; I am going to subtract 11x over there, so I get -2x = -16; x = +8--there is my x.
00:21:21.300 --> 00:21:45.600
Then, I have to look for all of my sides: AB is going to be 9 times 8 plus 12; that is 72 + 12, so AB is equal to 84.
00:21:45.600 --> 00:22:00.100
AC is 11 times 8, minus 4; that is 88 - 4, which is 84; that is AC.
00:22:00.100 --> 00:22:03.600
And notice how they are the same; they have to be the same, because that is the whole point.
00:22:03.600 --> 00:22:09.300
They are isosceles; these are congruent; we make them equal to each other so that they will be the same.
00:22:09.300 --> 00:22:26.800
And then, BC is 2 times 8 plus 10, so this is 16 + 10, so BC is 26.
00:22:26.800 --> 00:22:30.700
And that is it for this problem.
00:22:30.700 --> 00:22:34.700
Use the distance formula to classify the triangle by its sides.
00:22:34.700 --> 00:22:44.600
Here is my triangle, ABC; and then, you are going to use the distance formula to find what kind of sides there are.
00:22:44.600 --> 00:22:51.700
So, if I find the distance of A to C, then I will find the length of this side.
00:22:51.700 --> 00:22:56.000
Then I can use the distance formula to find the length of that side, and again do the same thing here.
00:22:56.000 --> 00:23:02.300
And then, you are going to compare those distances of the three sides.
00:23:02.300 --> 00:23:06.600
That means I will have to use the distance formula three times, for each of the sides.
00:23:06.600 --> 00:23:09.000
There is A, and there is B, and there is C.
00:23:09.000 --> 00:23:45.900
Now, A (let me just write it out) is at point...here is -2...1, 2, 3; B is at (-1,3); and C is at...here is 1, 2, 3...-1.
00:23:45.900 --> 00:23:50.900
Let's find AB first; it doesn't matter which one you find first.
00:23:50.900 --> 00:24:00.900
AB: I am going to use points A and B, these two; this is going to be x₂, and then I will just write out the distance formula again right here.
00:24:00.900 --> 00:24:12.700
All you do is subtract the x's, square that number, and add it to that number.
00:24:12.700 --> 00:24:41.700
AB is (-2 - -1)² + (-3 - 3)²; -2...minus a negative is the same thing as plus,
00:24:41.700 --> 00:25:02.800
so that is going to be -1 squared, is 1, plus -3 - 3, is -6, squared is +36; so this is going to be √37; there is AB.
00:25:02.800 --> 00:25:23.300
And then, BC is B and C: it is (-1 - 3)² + (3 - -1)².
00:25:23.300 --> 00:25:44.100
And then, -1 - 3 is -4, squared is 16; plus...this is going to be 3 + 1; that is 4; this is 16; and then, that is the square root of 32.
00:25:44.100 --> 00:25:51.400
This can be simplified; remember from the last lesson: if you want to simplify radicals,
00:25:51.400 --> 00:25:57.200
square roots, then you have to write this down; you can do a factor tree.
00:25:57.200 --> 00:26:03.000
This is going to be 16, and this is going to be 2; or you can just do 8 and 4; it doesn't matter.
00:26:03.000 --> 00:26:07.900
Circle it if it is prime; now, 16 here is a perfect square, so I can just go ahead and do that.
00:26:07.900 --> 00:26:17.900
But just to show you: it is going to be 8 and 2 (or 4 and 4; it doesn't matter), 4 and 2, 2 and 2.
00:26:17.900 --> 00:26:29.200
Whenever you have two of the same number, it is going to come out of the radical a single time.
00:26:29.200 --> 00:26:34.900
So then, as long as there are two of the same number, it comes out as one.
00:26:34.900 --> 00:26:43.800
So then, that comes out as a 2; these come out as a 2; and then, this one is still left.
00:26:43.800 --> 00:26:56.100
Whatever is left has to stay inside, and then, whatever came out (2 came out here, and 2 came out here)...that becomes 4√2.
00:26:56.100 --> 00:27:05.700
And then again, you know that is 16 times 2; we know that because it is 16 + 16; and then, just do that.
00:27:05.700 --> 00:27:27.200
That is BC; and then, AC is going to be (-2 - 3)² + (-3 - -1)².
00:27:27.200 --> 00:27:46.000
So, this is going to be -5, squared is 25, plus...this is going to be -2, because it is going to be plus; -2 squared is 4; so this is...
00:27:46.000 --> 00:27:55.100
OK, let me just write it here: AC = √29.
00:27:55.100 --> 00:28:00.400
And you can't simplify that any more.
00:28:00.400 --> 00:28:06.900
Now that I finished my distance formula, applying the distance formula to each of the sides,
00:28:06.900 --> 00:28:15.800
AC was √29; BC is 4√2; and then, AB is √37.
00:28:15.800 --> 00:28:24.600
Now, see how all three sides are different: the whole point is to classify the triangle by its sides.
00:28:24.600 --> 00:28:29.400
That means that it is either going to be a scalene, an isosceles, or an equilateral triangle.
00:28:29.400 --> 00:28:40.900
Since all three are different, we know that this is a scalene triangle; and that is your answer.
00:28:40.900 --> 00:28:43.000
That is it for this lesson; thank you for watching Educator.com.