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## High school geometry

### Course: High school geometry > Unit 2

Lesson 1: Rigid transformations overview- Getting ready for transformation properties
- Finding measures using rigid transformations
- Find measures using rigid transformations
- Rigid transformations: preserved properties
- Rigid transformations: preserved properties
- Mapping shapes
- Mapping shapes

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# Mapping shapes

Let's find the right sequence of rigid transformations (like rotations, translations, and reflections) to map one triangle onto another. Different sequences can work, but order matters. So, it's important to test each one to see if it maps the triangles correctly.

## Want to join the conversation?

- When I do the "Mapping Shapes" exercise, often times I have it matching up exactly with the shape,
**except**for the fact that it seems that the shape is slightly smaller than the one one the graph originally. It doesn't matter what the shape is, it always happened, and it's leaving me really frustrated and angry at myself. Can anyone help me with this issue please?(31 votes)- This may be because two of your sides almost look alike, but are not, as in a quadrilateral with two sides differing by one unit.

hope this helps with people who are reading this and about to do the practice.(8 votes)

- With the second example, why couldn't Sal just reflect the shape, instead of transforming and then reflecting?(20 votes)
- What does "a translation along the directed line segment HD" mean..Sal didn't teach about this kind of problem on mapping shapes.(25 votes)
- Nicholas Hood-Daniel gave the answer in the comments though so full credits to him:

it means putting point h to point d from one shape to the other and just draw the rest of the points in the same configuration as the pre image(0 votes)

- the comment section on khan academy is like mini-twitter(21 votes)
- We're told that triangles, let's see, we have triangle PQR and triangle ABC are congruent. The side length of each square on the grid is one unit. So each of these is one unit. Which of the following sequences of transformations maps triangle PQR onto triangle ABC? So we have four different sequences of transformations, and so why don't you pause this video and figure out which of these actually does map triangle PQR, so this is PQR, onto ABC, and it could be more than one of these. So pause this video and have a go at that. All right, now let's do this together. So let's first think about Sequence A, and I will do Sequence A in this purple color. So remember, we're starting with triangle PQR. So first it says a rotation 90 degrees about the point, about the point R, so let's do that. And then we'll do the rest of this sequence. So if we rotate this 90 degrees, so one way to think about it is, a line like that is then going to be like that. So we're going to go like that. And so R is going to stay where it is. You're rotating about it. But P is now going to be right over here. One way to think about it is to go from R to P, we went down one and three to the right. Now when you do the rotation, you're going to go to the right one and then up three. So P is going to be there and you can see that. That's the rotation. So that side will look like this. So that is P. And then Q is going to go right over here. It's going to, once again, also do a 90-degree rotation about R. And so after you do the 90-degree rotation, PQR is going to look like this. So that is Q. So we've done that first part. Then a translation six units to the left and seven units up. So each of these points are going to go six units to the left and seven up. So if we take point P, six to the left, one, two, three, four, five, six, seven units up, one, two, three, four, five, six, seven. It'll put it right over there, so that is point P. If we take point R, we take six units to the left, one, two, three, four, five, six, seven up, one, two, three, four, five, six, seven. It gets us right over there. And then point Q, if we go six units to the left, one, two, three, four, five, six, seven up is one, two, three, four, five, six, seven. Puts us right over there. So this looks like it worked. Sequence A is good. It maps PQR onto ABC. This last one isn't an R, this is a Q right over here. So that worked, Sequence A. Now let's work on Sequence B. I'll do this in different color. A translation eight units to the left and three up, so let's do that first. So if we take point Q eight to the left and three up, one, two, three, four, five, six, seven, eight, three up, one, two, three. So this'll be my red Q for now. And now if I do this point R, one, two, three, four, five, six, seven, eight. Let me make sure I did that right. One, two, three, four, five, six, seven, eight, three up, one, two, three. So my new R is going to be there. And then last but not least, point P, eight to the left, one, two, three, four, five, six, seven, eight, three up, one, two, three, goes right there. So just that translation will get us to this point. It'll get us to that point, so we're clearly not done mapping yet, but there's more transformation to be done. So it looks something like that. Says then a reflection over the horizontal line through point A. So point A is right over here. The horizontal line is right like that. So if I were to reflect, point A wouldn't change. Point R right now is three below that horizontal line. Point R will then be three above that horizontal line. So point R will then go right over there. Just from that, I can see that this sequence of transformations is not going to work. It's putting R in the wrong place. So I'm going to rule out Sequence B. Sequence C, let me do that with another color. I don't know, I will do it with this orange color. A reflection over the vertical point through point Q. Sorry, a reflection over the vertical line through point Q. So let me do that. So the vertical line through point Q looks like this, just gonna draw that vertical line. So if you reflect it, Q is going to be, it's going to stay in place. R is one to the right of that, so now it's going to be one to the left once you do the reflection. And point P is four to the right, so now it's gonna be four to the left. One, two, three, four, so P is going to be there after the reflection. And so it's going to look something like this after that first transformation, and this is getting a little bit messy. But this is what you'll probably have to go through as well, so I'll go through it with you. All right, so we did that first part, the reflection. Then a translation four to the left and seven units up. So four to the left and seven up. So let me try that. So four to the left, one, two, three, four, seven up, one, two, three, four, five, six, seven. So it's putting Q right over here. I'm already suspicious of it because Sequence A worked where we put P right over there. So I'm already suspicious of this, but let's keep trying. So four to the left and seven up, one, two, three, four, seven up, one, two, three, four, five, six, seven. So R is going to the same place as Sequence A put it. And then point P, one, two, three, four, one, two, three, four, five, six, seven. Actually, it worked. So it works because this is actually an isosceles triangle. And so this one actually worked out. We were able to map PQR onto ABC with Sequence C. So I like, I like this one as well. And then last but not least, let's try Sequence D. I'll do that in black so that we can see it. So first we do a translation eight units to the left and three up. Eight to the left and three up. So we'll start here. One, two, three, four, five, six, seven, eight, three up, one, two, three. So I'll put my black Q right over there. So eight to the left, one, two, three, four, five, six, seven, eight, three up, one, two, three. I'll put my black R right over there. It's actually exactly what we did in Sequence B the first time. So P is going to show up right over there. So after that translation, sequence, that first translation in Sequence D gets us right over there. Then it says a rotation -270 degrees about point A. So this is point A right over here. And -270 degrees, it's negative, so it's going to go clockwise. And let's see, 180 degrees, let's say if we were to take this line right over here, if we were to go 180 degrees, it would go to this line like that. And then if you were to go another 90 degrees, it actually does look like it would, it would map onto that. So this is actually looking pretty good. If you were to, this line right over here will then, if you go -270 degrees, will map onto this right over here, and then that point R will kinda go along for the ride is one way to think about it, and so it'll go right over there as well. So I'm actually liking Sequence D as well. So all of these work except for Sequence B.(0 votes)

*I really can't grasp the concept of this, can someon reply with the basics?***I will be very much thankful**(7 votes)- This might be a little late, and I am not sure if this will help, but I will try.(Also, if they are any grammatical mistakes, please forgive that.)

So, first to understand this, you must know and understand the concept of 4 transformations: Translations, Rotations, Dilations, and Reflections. Translation is basically moving an image by a certain number of x and y coordinates, without changing anything. Rotations are when you rotate a shape by some amount of degree, usually 90, 180, or 270 degrees. Dilations are when you shrink or enlarge a shape. Finally, a reflection is basically when you reflect an image or a coordinate point across a line. This line could be x-axis, y-axis, or any other line the problem specifies.

In this video, Sal, the one who is talking in the video, is trying to find which sequences could transform the image PQR to ABC. To do that, multiple transformations were applied. Trying to find which transformations were applied is what all this video and the concept is trying to explain and find. Also, note that there is more than one way to apply the transformation from PQR to ABC.

I hope this helped :)(17 votes)

- Can u please teach it a little slower....please😁(14 votes)
- What is a translation over a directed line segment?(6 votes)
- A directed line segment is a segment that has not only a length (the distance between its endpoints), but also a direction (which means that it starts at one of its endpoints and goes in the direction of the other endpoint).

For example, directed line segment 𝐴𝐵 starts at 𝐴 and ends at 𝐵 (not the other way around).

A translation over directed line segment 𝐴𝐵 means that we are translating a point from its current location at 𝑃 to its new location at 𝑄, such that directed line segment 𝑃𝑄 has the same distance and direction as 𝐴𝐵.(15 votes)

- I do not have a good imagination, so I find this exercise pretty much impossible(10 votes)
- Like Darran said, use graph paper to Practice practice practice. Imagination is just like any other skill, it can be trained (unless you have aphantasia). I too am not a visual learner, but with practice it comes.(1 vote)

- Seriously, this is the hardest thing on here. Taken the quiz 5 times and hand written on graph paper and still cannot pass(10 votes)
- Don't try to copy it on graph paper: use the pen tool right on the screen (bottom left of the quiz screen). Don't try and skip steps: draw every step. If you passed the other quizzes then you already have the skills to do this, so don't stress out: this quiz just puts two skills together in each question.(2 votes)

- I dont get any of this I'm going to sue.(10 votes)

## Video transcript

- [Instructor] We're told that triangles, let's see, we have triangle PQR and triangle ABC are congruent. The side length of each square
on the grid is one unit. So each of these is one unit. Which of the following
sequences of transformations maps triangle PQR onto triangle ABC? So we have four different
sequences of transformations, and so why don't you pause this video and figure out which of
these actually does map triangle PQR, so this is PQR, onto ABC, and it could be
more than one of these. So pause this video and have a go at that. All right, now let's do this together. So let's first think about Sequence A, and I will do Sequence
A in this purple color. So remember, we're
starting with triangle PQR. So first it says a rotation 90 degrees about the point, about the
point R, so let's do that. And then we'll do the
rest of this sequence. So if we rotate this 90 degrees, so one way to think about it is, a line like that is then
going to be like that. So we're going to go like that. And so R is going to stay where it is. You're rotating about it. But P is now going to be right over here. One way to think about
it is to go from R to P, we went down one and three to the right. Now when you do the rotation, you're going to go to the
right one and then up three. So P is going to be there
and you can see that. That's the rotation. So that side will look like this. So that is P. And then Q is going to go right over here. It's going to, once again, also do a 90-degree rotation about R. And so after you do
the 90-degree rotation, PQR is going to look like this. So that is Q. So we've done that first part. Then a translation six units to the left and seven units up. So each of these points
are going to go six units to the left and seven up. So if we take point P, six to the left, one, two,
three, four, five, six, seven units up, one, two,
three, four, five, six, seven. It'll put it right over
there, so that is point P. If we take point R, we
take six units to the left, one, two, three, four, five, six, seven up, one, two, three,
four, five, six, seven. It gets us right over there. And then point Q, if we
go six units to the left, one, two, three, four, five, six, seven up is one, two, three,
four, five, six, seven. Puts us right over there. So this looks like it worked. Sequence A is good. It maps PQR onto ABC. This last one isn't an R,
this is a Q right over here. So that worked, Sequence A. Now let's work on Sequence B. I'll do this in different color. A translation eight units to the left and three up, so let's do that first. So if we take point Q eight
to the left and three up, one, two, three, four, five, six, seven, eight, three up, one, two, three. So this'll be my red Q for now. And now if I do this point R, one, two, three, four, five, six, seven, eight. Let me make sure I did that right. One, two, three, four, five,
six, seven, eight, three up, one, two, three. So my new R is going to be there. And then last but not least, point P, eight to the left, one, two, three, four, five, six, seven, eight, three up, one, two,
three, goes right there. So just that translation will get us to this point. It'll get us to that point, so we're clearly not done mapping yet, but there's more
transformation to be done. So it looks something like that. Says then a reflection
over the horizontal line through point A. So point A is right over here. The horizontal line is right like that. So if I were to reflect,
point A wouldn't change. Point R right now is three
below that horizontal line. Point R will then be three
above that horizontal line. So point R will then go right over there. Just from that, I can
see that this sequence of transformations is not going to work. It's putting R in the wrong place. So I'm going to rule out Sequence B. Sequence C, let me do
that with another color. I don't know, I will do
it with this orange color. A reflection over the vertical
point through point Q. Sorry, a reflection over the
vertical line through point Q. So let me do that. So the vertical line through
point Q looks like this, just gonna draw that vertical line. So if you reflect it, Q is going to be, it's going to stay in place. R is one to the right of that, so now it's going to be one to the left once you do the reflection. And point P is four to the right, so now it's gonna be four to the left. One, two, three, four,
so P is going to be there after the reflection. And so it's going to
look something like this after that first transformation, and this is getting a little bit messy. But this is what you'll probably
have to go through as well, so I'll go through it with you. All right, so we did that
first part, the reflection. Then a translation four to
the left and seven units up. So four to the left and seven up. So let me try that. So four to the left,
one, two, three, four, seven up, one, two, three,
four, five, six, seven. So it's putting Q right over here. I'm already suspicious of
it because Sequence A worked where we put P right over there. So I'm already suspicious of
this, but let's keep trying. So four to the left and seven up, one, two, three, four, seven up, one, two, three, four, five, six, seven. So R is going to the same
place as Sequence A put it. And then point P, one, two, three, four, one, two, three, four, five, six, seven. Actually, it worked. So it works because this is
actually an isosceles triangle. And so this one actually worked out. We were able to map PQR
onto ABC with Sequence C. So I like, I like this one as well. And then last but not
least, let's try Sequence D. I'll do that in black
so that we can see it. So first we do a translation
eight units to the left and three up. Eight to the left and three up. So we'll start here. One, two, three, four, five, six, seven, eight, three up, one, two, three. So I'll put my black Q right over there. So eight to the left,
one, two, three, four, five, six, seven, eight, three up, one, two, three. I'll put my black R right over there. It's actually exactly
what we did in Sequence B the first time. So P is going to show up right over there. So after that translation, sequence, that first
translation in Sequence D gets us right over there. Then it says a rotation
-270 degrees about point A. So this is point A right over here. And -270 degrees, it's negative, so it's going to go clockwise. And let's see, 180 degrees,
let's say if we were to take this line right over here, if we were to go 180 degrees, it would go to this line like that. And then if you were to
go another 90 degrees, it actually does look like it would, it would map onto that. So this is actually looking pretty good. If you were to, this line
right over here will then, if you go -270 degrees, will
map onto this right over here, and then that point R will
kinda go along for the ride is one way to think about it, and so it'll go right over there as well. So I'm actually liking Sequence D as well. So all of these work
except for Sequence B.