Geometry Special Right Triangles

Welcome back to Educator.com.0000

For this next lesson, we are going to go over special right triangles.0002

Now, we are still on right triangles; so make sure that all of these are only being used for right triangles.0006

That includes the Pythagorean theorem; that includes the altitude to find the geometric mean, and this one, which is special right triangles.0015

Now, we have two types of special right triangles.0026

Now, what makes them special: it is a right triangle, but it is special because of the type of right triangle that it is.0029

If it is a 45-45-90 degree triangle, meaning that the angle is 45, the other angle is 45, and then, of course,0040

we have the 90-degree right angle, then there is a shortcut to finding the other sides of the triangle.0049

That is why it is special--because there is a shortcut.0059

Again, there are two of them: if you have any other type of triangle, besides these two special right triangles,0062

then you are going to have to use one of the other methods, depending on what you are trying to find.0069

For a 45-45-90 degree triangle, we don't have to use the Pythagorean theorem;0075

we don't have to use geometric mean and some of the other concepts that we are going to go over in the next lesson.0080

For the 45-45-90 triangle, we know that the hypotenuse is going to be √2 times [as long as the leg].0091

Now, we know that a triangle has three sides and three angles.0102

For each angle, there is a side that is corresponding with it; it is always the angle and the side opposite that angle that kind of pair up.0109

They are kind of like a couple; they pair up, so they are related.0123

We have...if this is triangle ABC, then I can say that the side opposite angle A...this...I can name that side a (lowercase a for the sides).0132

And then, here, this side AC is going to be corresponding with this angle B; so it is as if that is b, because this angle and this side go together.0148

And then, for this angle C and the hypotenuse, they are kind of related, too.0162

Keep that in mind--that is very important to remember: the angle and the side opposite go together.0169

In a 45-45-90 triangle, how do I know that this is a 45-45-90 degree triangle?0179

It is only given that this is 45 and this is 90, but I know that all three angles of a triangle have to add up to 180.0186

So, if this is given to me as 45, and this is given, then I know that the third angle has to be 45, as well.0194

45-45-90: the side opposite the 45-degree angle, we are going to name as n; so then, these two are going to go together.0209

So then, this one is going to be n; now, I am going to erase this, so that you don't get it confused with all of these variables.0229

The side opposite the 45 is n; well, if the side opposite this 45 is n, then the side opposite this 45 also has to be n,0240

because remember: this is an isosceles triangle, and because of the base angles theorem,0250

if these angles are the same, then those sides opposite have to be the same.0255

If this is n, then this has to also be n.0262

And then, the hypotenuse, then, the side opposite the 90-degree angle, is going to be √2 times that side.0268

So, if this is n, then this is going to be √2n, or let me write n√2, n times √2.0281

Just keep this in mind: 45-45-90 is going to be n, n, n√2.0300

Let's say that this CB is, let's say, 4; then this side right here is going to be 4, and AB is going to be 4√2,0311

because it is √2 times n; n is 4; 4 times √2 is 4√2.0334

That is the shortcut for a 45-45-90 degree triangle: n, n, n√2.0340

Here, they give you one of the sides, and you have to find n.0355

Now, for this one, this one is a little bit different; why?--because we know that, if this is n, what is this? n.0363

What about the side opposite the 90?--45-45-90--it is going to be n, n (for the side opposite this 45),0377

and then, for this one, it is going to be n√2; the side opposite the 45 is n; this is also 45,0389

so that is going to be n; what about the side opposite the 90?--it is going to be n√2.0399

We don't know n; we want to find n.0407

They give us this one right here, so we know that this is 10.0409

What I can do, since I know that this is equal to n√2, which is 10: I can just make that into an equation.0416

n√2, this side right here, is 10; I can solve for n.0428

So then, how do I do this? I would have to divide by √2, so n is equal to 10/√2.0436

Remember: I can't have a radical in my denominator, so I have to rationalize it.0449

I have to multiply this by √2/√2, because √2/√2 makes one; anything over itself is 1.0455

So, when I multiply this out, it is going to be 10√2 over...what is √2 times √2? 2.0466

And then, this will become 5√2; so n is equal to 5√2.0476

That means that this is 5√2, and this is 5√2.0485

Again, just make sure that you understand this; this is n; this is n; then, this would be n√2.0492

Whatever side they give you, as long as they give you one side--you can find the remaining sides using the shortcut, using this.0503

But you just have to make sure that you are going to make that measure equal to one of these, depending on what side they give you.0511

If they give you one of the sides opposite the 45, then that is n.0519

You are going to make that equal to n.0525

If they give you the hypotenuse, you are going to make that equal to n√2, and then you are going to solve for n by making it equal to each other.0527

So, on to the next special right triangle: that is a 30-60-90 triangle.0539

If you have a triangle whose angle measures are 30, 60, and 90, then we can also use another shortcut.0547

Here, they don't have to give you that this is 60, as long as they give you that one of them is a right triangle, and one of them...0563

this is either 60, or this is 30; then you know that it is a 30-60-90 triangle,0570

because, if you are going to just subtract it from 180, you are going to get that remaining angle measure.0574

And it is going to become either 30 or 60.0581

So then, a 30-60-90 triangle: the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.0583

The shorter leg is the side opposite the 30-degree angle: 30, 60, 90.0599

This is a little bit trickier; the side opposite the 30...we are going to name that n.0610

Now, here, this 60-degree one, the longer leg, is going to be n√3.0622

Now, be careful here, because the 2n, the one that is twice as long as the shorter leg, is the hypotenuse, 2n.0632

This is probably where students make the biggest mistake, because 60 is two times 30,0645

so they automatically assume that, if this is n, then this has to be 2n.0655

That is not the case; this is n, n√3, and then this one, the one opposite the 90, the hypotenuse, is 2 times n.0659

So, be very careful with that: n, n√3, and 2n.0670

Let's write the 45-45-90; this one is n, n, n√2; this is n√3 for the 60 one.0674

So, be careful; this is n√2; this is n√3; that is also another common mistake.0689

Instead of writing n√3, they write n√2; or with this one, instead of writing n√2, they write 3; so just be careful with that.0695

Let's say that this one right here is 4; then, the one opposite the 60 would be n√3, so that would be 4 times √3, 4√3.0709

And the hypotenuse is going to be 2 times that side, n; so 2 times n is 8.0725

So, if it is 4 here, then it is going to be 4√3 here, and it is going to be 8 there.0734

Let's do one example of this: again, let's write it out: a 30-60-90 triangle is going to be n; this one is not 2n, but n√3; this one is going to be 2n.0745

Here, this is n; this one...oh, this is 30 degrees; this is 60 degrees...the one opposite the 30 is n, which is what they want you to find.0771

The one opposite the 60 is 12; remember, again, as long as they give you the measure of one side,0785

you can find the remaining sides, because we have a shortcut here.0792

Here, this one opposite 60 is 12; that means that this is 12, right here.0799

And then, the hypotenuse we don't know; all we know is that that is that, and then we know that 12 is equal to...0810

this is n; this is n√3; this is 2n; we know that n√3 is equal to 12.0822

So, all you have to do is make them equal to each other: n√3 is equal to 12.0830

To solve for n, divide by √3; n is equal to 12/√3.0841

Again, we have to simplify this by rationalizing the denominator.0852

√3/√3 becomes 1; we have to do that, because we need to get rid of this radical; it will become 12√3/3, so n is 4√3.0858

n is 4√3; that means that this one here is 4√3.0878

And then, what about the hypotenuse? It is 2 times that number, so it is 2 times 4√3, which is 8√3.0885

So again, whatever side they give you, just make it equal to that shortcut; and then, you just solve out for n.0898

Let's do a few examples: Find the values of x and y.0908

Here is a 45-45-90 degree triangle; that means that we can use our shortcut.0914

They gave us one side; the 45-45-90 one is n, n, n√2.0920

What do they give us? They give us the one opposite the 45, so they give us this one right here.0935

This one is 15; so what are the other sides?0943

This one here, opposite the 45, is also 15.0950

And then, how about this one? This would be 15 times √2, which is 15√2.0957

So again, this one is given, the side opposite the 45.0969

Then, we can find our other two sides, because if 15 is n, then what is n here? 15 is n.0976

Then, n√2 would be 15√2.0985

Here we have a 30-60-90 degree triangle; 30 is n; this one is not 2n, but n√3; and this one is 2n.0993

Which one is given? Let's see, this one, the hypotenuse, is given, and that is the one opposite the right angle, right here.1021

Here, this is going to be 22; that means I can just make that equal to each other.1031

Since this is 2n, 2n is equal to 22; so n is 11.1038

Which one is n? Isn't that here, the one opposite the 30?1054

So, this is 11, and then this one is going to be 11√3.1060

The side opposite 30, x, is 11; and then, y is the side opposite 60, the long leg, so it is 11√3.1068

Let's write out x and y for this one, too.1084

If this side is 15, then x is 15; what is y? x√2...so it is 15√2.1086

The next one, Example 2: Find the values of x and y.1102

OK, so here, this is actually a square; and this is a diagonal of the square, so that makes this angle a 45-degree angle.1106

Automatically, I know that x is 45 degrees, which means that this would also be 45; here is a right angle; here is 45.1124

This right here now becomes a 45-45-90 degree triangle.1139

45, 45, 90: the side opposite the 45 is n, so this is going to be n; the side opposite 90 is going to be n√2.1146

Now, what are we given? We are given the hypotenuse, 16, so that is this right here; the side opposite the 90 is 16.1163

That means that I am going to make this equal to that, so n√2 = 16.1178

Divide by √2; n is equal to 16/√2; multiply this by √2/√2; 16√2/2--that makes n 8√2, which is, in this case, y.1190

In this problem, it is y, not n.1216

So then, here, y is equal to 8√2; those are my answers.1219

The next one: let's see, 30-60-90...how do I know that?--because look at that triangle right there: 30, 60, 90.1232

Let's see, they want to know this right here and this whole side right here.1251

If 30 is n, 60 will be n√3, and 90 will be 2n.1267

From this triangle, this right here is also 60, which would make this 30.1278

Now, it is the same thing to find this right here, to use this triangle, or to use this triangle, because it is still 30-60-90.1296

Here is the hypotenuse; here is x, what we are looking for; but what is given?1309

Look at what is given: they have to give you at least one side, so in this problem, they give you the side opposite the 60, the long leg.1315

This one is 10.5; so we are going to have to solve that out and make them equal to each other: 10.5,1326

then divide...n is equal to 10.5 divided by √3.1339

Now, this is decimals, so you want to use your calculator.1359

n is 10.5√3, over 3; and again, use your calculator.1369

Or if you want to leave it as a fraction, then you can just change this 10.5; 10.5, we know, is 10 and 1/2.1378

If you change that to an improper fraction, it becomes 21/2; so then, this can be the same thing as 21/2 √3/3.1392

Now, this is also this, which is 21√3/6.1413

All I did was just to change 10.5 into a fraction, which became 21/2.1435

And then, I multiplied it to √3, so it became 21√3/2.1441

And then, you just divide it by 3; and when you divide, it just becomes 21√3/6.1446

You can leave it like that, as long as you don't have a radical in the denominator,1455

and as long as these numbers don't simplify; then that would be the answer.1458

Otherwise, you can just change it to decimals; because this is given in decimals, you can go ahead and use your calculator and find the decimal of that.1465

I don't have a calculator that I can use to give you the decimal answer, so you would have to just punch in...1473

you could do 21 times √3 over 6, divided by 6, or do 21 divided by 6, and multiply that by √3.1481

I am just going to say that this n, which is the side opposite here, is 21√3/6.1492

Now, they are not asking for this side right here; they are asking for this side, which is the side opposite the 90, the hypotenuse.1504

So, I need to multiply that by 2; 2 times this whole thing is going to be (I'll do it right here) 21√3/6.1513

Now, I can just go ahead and simplify this, or I can multiply it out; it becomes 42√3/6, and then you have to simplify.1527

Or just change this to a 3; so it becomes 21√3/3, which would then become 7√3.1535

x is 7√3, and then y is also the same answer.1557

Now, if this right here was 21√3/6--this was the answer to this--why is it just 2 times that amount?1576

Now, if you look here, this is 60, this is 60, and this is 60.1592

That makes this an equilateral triangle--equiangular, and therefore equilateral.1597

Whatever this side becomes, this side is, also.1602

Now, you can also think of it as these two triangles being the same; so if this is 21√3/6, then this is 21...1606

this actually simplifies, so this becomes 2, and then this becomes 7; so it becomes 7√3/2.1621

This is also 7√3/2; so if you just add them together (or you can just do 7√3/2 times 2, and that is the same thing), y is 7√3, the same answer.1638

The third example: The perimeter of an equilateral triangle is 27 inches; find the length of an altitude of the triangle.1662

We have an equilateral triangle; the perimeter is 27 inches.1674

So, if it is equilateral, that means I know that this side, this side, and this side are all the same,1684

which would make each side 9, because it is 27 divided by 3.1690

Find the length of an altitude of the triangle.1695

If it is an equilateral triangle, then it is also equiangular, which means that this is 60, 60, and 60,1699

because all three angles of a triangle have to add up to 180, so it is 180 divided by the three angles.1709

I need to find an altitude; an altitude is that right there.1717

If this is 60 and this is 90, then this has to be 30; that means we are using a 30-60-90 triangle.1727

Then, here is 30; that is n; this is n√3; here, this is 2n.1739

What is given to us? The hypotenuse, the side opposite the 90--this one is 9.1748

So then, I just need to make those two equal to each other; divide by 2; n is...I can leave it as 9/2, or 4.5.1757

And what do they want us to find?1771

They want us to find this right here, the altitude.1772

What did we find? We found the side opposite the 30; this one is 4.5.1776

How do you find the one opposite the 60?1783

It is n times √3, so we have to do x = 4.5 times √3, so it is 4.5√3; that is the altitude.1786

If you want to leave it as a fraction, you can say that it is 9/2, times √3; it is going to be 9√3/2--that is also the answer.1809

The next one: the length of a diagonal of a rectangle is 15 centimeters long and intersects the vertex to make a 60-degree angle.1827

Find the perimeter of the rectangle.1837

The rectangle...the length of a diagonal...I need a diagonal; this is 15; these are our right angles.1844

This intersects the vertex to make a 60-degree angle.1859

This whole thing is 90; that means that, if one of these is 60, the other one has to be 30.1864

So, I want to make the bigger one 60; this is going to be 30.1870

Find the perimeter of the rectangle: if this is 30, then this is 60; it has to be a 30-60-90 degree triangle.1876

30 + 60 + 90 has to be 180; find the perimeter.1885

OK, if I am going to find the perimeter, that means that I have to find the measure of all of my sides.1890

I have to find this side, and I have to find this side.1896

With 30-60-90, this is n, n√3, 2n.1900

They give us the side opposite the 90, which is this, 15; make them equal to each other.1920

n = 15/2; that is this one right here, 15/2, the side opposite 30.1930

What is this? It is 15/2, and then what is the one opposite the 60?1941

It is n√3, so it is 15/2 times √3, or 15√3/2.1954

To find the perimeter, I have to add up all of the sides, so it is going to be this side times 2,1964

so the perimeter is going to be 2(15√3/2) + 2 times this side, 15/2...1972

Again, perimeter is just this plus this; it is 2 times this side, plus 2 times this.1992

So then, this becomes (I'll just cancel that out) 15√3 + 15.2002

And I cannot add these two up, because this one has √3, so this would just be the answer.2012

I'll just write it out here: the perimeter equals 15√3 + 15, and that would be the answer.2019

Next, Example 4: The regular hexagon is made up of 6 equilateral triangles; find AC.2031

A regular hexagon..."regular" means that it is equilateral and that it is equiangular.2043

Here is a hexagon, because it has six sides; and "regular"--each of these sides is congruent.2049

I know that, if it is regular, then that means that it is made up of six congruent triangles.2061

Each of these triangles is congruent; that means that this side is going to be 12; these angles are going to be 60,2078

because again, "equilateral triangles" would also mean that they are equiangular,2096

and that means that all of the angles have to be 60 (because together it has to be 180, and then you just divide it by 3).2102

We have to find AC; here, this is perpendicular...so then, if we look at this triangle right here,2110

this one with this one and this, this is 60; what would that make this?2120

This would be 30, because it is these two parts; so here, we have a 30-60-90 triangle.2132

Now, if I find this right here, then I can just multiply it by 2 to get AC.2139

I know that this side right here is 12; I am going to be using a 30-60-90 triangle, n, n√3, and 2n.2149

From here, look at that triangle right there; what side is given to me?--the hypotenuse, which is this: 2n = 12, so n is 6.2165

That is this right here; so this is 6; what is this right here, then?2184

That would be n√3, which is 6√3.2189

I need to find AC; well, if this is 6√3, that means that this has to be 6√3, also.2195

So, AC then would be 2 times 6√3, so AC is 12√3.2202

Just to go over what we just did: this hexagon is made up of 6 equilateral triangles,2222

so I just split it up into 6 triangles; I am going to focus on this triangle right here, and then half of that; that way, I can find this right here.2229

And then, since I know that I have a 30-60-90 triangle, I can use my shortcut.2242

My hypotenuse is 12, so I made 12 equal to 2n, solved for n, and then found this side, which is 6,2247

and then this side, which is 6√3; this is what I need.2258

I multiply that by 2, so I get 12√3; so AC, which is what they want me to find, is 12√3.2263

That is it for this lesson; thank you for watching Educator.com.2274