SM2 10.2: Solving Similar Triangles, Triple Triangles v1

Added: 2026-04-29

[00:00:01]hello and welcome. So given the triangle  information below, solve for each variable.

[00:00:06]Write your answers as a decimal value to the  third decimal place. All right, keep in mind diagrams not to scale. All right, so first off we  have on the left side some lengths. On the right side we have a triangle. Now when we're looking  at this there's a couple of things to note. We're looking at similar triangles specifically, okay?  For similar triangles you're going to be creating proportions and proportions are written of each  triangle that we have. Well how many triangles do you see right here? Three. We have one is this  little triangle, two is the medium triangle, and then we have the whole thing itself is also  a triangle. So you have three similar triangles here. So first thing I would recommend is we're  going to put the information on the diagram, right? But it's also going to be helpful to write  three separate triangles like down below in your work. Make sure you can tell each triangle apart.  Write them all in the same orientation so it'll help you a lot when it comes to writing those  proportions. All right, so first off let's just let's get all the information down. So AC this  section right here we're told is 15, okay? And you'll notice that that is the hypotenuse across  from the right angle. That'll be this side right there. That is 15 on your diagram, okay? And now  on the next piece we're told AD which is this little section right there is going to be five.  And then that because your right angle also right here so that's going to be the short side of your  smallest triangle. So right here is right there so that is five, okay? And then finish it off. If  we know that's 15 and five then this clearly has to be 10 right there. And we're also told that  DC that section right there is 10 so huzzah, okay? And then from there that'll just fill  in the last piece of information right. This is your medium triangle and it's going to be  the longer side, all right? Which means that's going to be that side right there, 10, okay? And  then from there we're going to be finding three pieces of information. Told to find BC which I'm  gonna call that uh x for now. So there's x which is going to be the hypotenuse. That side right  there. And it's also going to be the longer side of the large triangle. So x is there. And then  we keep going. BD I'm going to call y just for the sake of calling it y. Why not? All right? And  that's going to be this little section. Oops let's not use green, right here, all right? Which will  be the short section of your middle triangle and the long side of your little one. So there's y  and there's y. All right? Last one I'm going to call that z and that'll be from AB and that will  be the section right there which will be the short side of your big triangle and the hypotenuse of  your little triangle. So that's z and that's z, okay? And now through all of that we have filled  in all of our information right there and it's just a matter of solving uh with proportions now,  okay? So pick a variable you want to solve for: the x, the y, or the z, okay? And then try setting  up proportions for whatever it is you pick. I'm just going to go down the list. I'm just going  to start with x and then go to y then to z. So in order to solve for this that's all right. So first  thing we want to do is we want to figure out all right where do we have X's. Which triangles? Which  of the three triangles do we see X's on? Well we see on the middle and the large and those are the  two triangles you're going to use for proportions, all right? So now you decide you want to go from  the small the medium triangle to the big one or the big one to the medium. I like going bigger to  smaller so I'm going to go big side. I'm going to start with x which will correspond to the same  thing on the other triangle. So x over 10. And that's why you want to set up your triangles in  the same orientation here is because it lets you just do that side to side comparison, all right?  And then same thing. Now you can use either y, z over y or 15 over x, all right? Now remember you  want to solve for x's so using the sides with x is what you want to do. So then we do 15 over x. Now  it is 15 over x not x over 15 because we used x before so it has to stay consistent: big to small,  big to small, okay? All right and that's it. Those are your proportions for x and now you are set  to solve for x, okay? Fractions are division: undo division with multiplication. I'm going to  multiply x on both sides and I'm also going to multiply 10 on both sides so that way it'll cancel  out the tens and the x's. And then you have x times x, x square right? And then 15 times 10 which is 150. Oops  no square root. 150, okay? So x squared equals 150. Now you can solve this a couple of ways  right but what I would recommend doing is you already have your x alone and it's squared. How do  you undo squares? Square root. Don't forget your plus/minus, right? But in this case think about  it we're talking about lengths. Can this be a negative number? No, lengths can't be negative. So  while there is technically your plus or minus you only need the positive square root, right? Now you  don't need to simplify this. Remember the problem said to write your answer as a decimal so you only  need square root of 150. Don't bother simplifying it. You don't need to, okay? And we want to go to  the third decimal place. That'll be 12.247 because the four behind will round down. So 12.247. Done.  x out of the way. I'm going to go a little bit faster when I solve for the other two. Let's  do the y's. You see y's in these two triangles specifically. These are all of your sides with  y's, okay? I'm doing big to small again so y over 5 equals 10 over y because staying consistent:  large to small, large to small. You want the sides with y's in there so that's why we grab those  ones, all right? Multiply by a y and a five both sides so the y's cancel on the right, fives on  the left and you get y squared equals 50. Take the square root. Technically there's plus minus but  you only need the plus so y equals square root of 50, okay? So then we come back. Square root of 50  we get 7.071. So we have 7.071 done, okay? Now we have that last but not least moment right which is  solving for z. You see a z in the first and last triangle. Here's a z. There's its corresponding  side. Here's a z. Here's its corresponding. So then set up your proportions: 15 over z big  to small equals z over 5 big to small, right?

[00:08:00]Multiply by five and z. All right? Fives cancel  there. z's cancel here and you're left with five times 15 which is 75 on the left equals z times  z. z squared. Undo squares with square roots. Only need the positive though. So z is the positive  square root of 75. So 8.660, okay? Honestly it's not too hard. It's not too difficult in terms  of algebra. The biggest thing that's going to trip a lot of you guys up is probably going  to be how to set up those triangles to get the corresponding sides down, all right? But that's  all there is to it. So thank you for watching.