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### Special Right Triangles

- In a 45°-45°-90° triangle, the hypotenuse is times as long as the leg
- In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg

### Special Right Triangles

If the measure of an acute angle is 45

^{o}in a right triangle, then the two legs are concrugent.

^{o}− 45

^{o}− 90

^{o}triangle, if the measure of the leg is 4, find the measure of the hypotenuse.

In a 30

^{o}− 60

^{o}− 90

^{o}triangle, the hypoenuse is 2 times as long as both legs.

In a 45

^{o}− 45

^{o}− 90

^{o}triangle, the hypotenuse is √2 times as long as the leg.

^{o}, mC = 30

^{o}, AB = 3, find BC and AC.

- This is a 30
^{o}− 60^{o}− 90^{o}triangle - BC = √3 AB = 3√3

^{o}, AB = 3, find BC and AC.

- This is a 45
^{o}− 45^{o}− 90^{o}triangle - AC = AB = 3

^{o}− 45

^{o}− 90

^{o}triangle, so the measures of three angles are 45

^{o}, 45

^{o}and 90

^{o}.

Isosceles triangle ABC, AB = AC, the altitude of the triangle ―AD is [(√3 )/2] times as long as ―BC , AD = 3, find the measure of ―AB .

- AD = [(√3 )/2]BC
- BD = [1/2]BC
- AD = √3 BD
- right VABD is a 30
^{o}− 60^{o}− 90^{o}triangle - BD = [AD/(√3 )] = [3/(√3 )] = √3

^{o}− 60

^{o}− 90

^{o}triangle, so the measures of three angles are 30

^{o}, 60

^{o}and 90

^{o}.

The perimeter of square ABCD is 8, find the length of ―AC .

- AB = [1/4] perimeter = [1/4]*8 = 2
- ∆ABC is a 45
^{o}− 45^{o}− 90^{o}triangle,

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Special Right Triangles

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- 45-45-90 Triangles 0:06
- Definition of 45-45-90 Triangles
- 45-45-90 Triangles 5:51
- Example: Find n
- 30-60-90 Triangles 8:59
- Definition of 30-60-90 Triangles
- 30-60-90 Triangles 12:25
- Example: Find n
- Extra Example 1: Special Right Triangles 15:08
- Extra Example 2: Special Right Triangles 18:22
- Extra Example 3: Word Problems & Special Triangles 27:40
- Extra Example 4: Hexagon & Special Triangles 33:51

### Geometry Online Course

### Transcription: 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

0 answers

Post by Denise Bermudez on March 11, 2015

hi!

I have a small question...in minute 14:29 why after you get n=12/the square root of 3 you get an answer of just a 3 on the bottom of 12 * square root of 3?

You always said that any number divided by itself is equal to 1. Why do you get just that 3 as a parcial answer?