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Course: Integrated math 1 > Unit 16
Lesson 3: Properties & definitions of transformationsDefining transformations
Given a description of the effect of a certain transformation, we determine whether that transformation is a translation, a rotation, or a reflection.
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Sal is starting to use the phrase "maps to itself" a lot and the verb 'map'. In other words, what does Point O maps to itself mean at3:00?(37 votes)
Mapping a point basically means to change the coordinates. Mapping is mostly used on the coordinate plane, where points can be transformed (or mapped) to another place. Now, to answer your initial question, "Point O maps to itself" means that when it is transformed using some kind of transformation (rigid or non-rigid), it maintains its original position on the coordinate plane. Hope this helps! :)(25 votes)
I feel like I don't understand english anymore when I read these kind of problems xD(38 votes)
What's a perpendicular bisector? Did I miss something in one of the videos or is that just new math language?(21 votes)
Break it into the two words:
Perpendicular means intersecting at right angles
Bisector means cutting into two equal parts (so cutting a line segment in half)
So perpendicular bisector would be a line or line segment that is perpendicular to a line segment and cuts that line segment in half(25 votes)
- What does Sal mean by 'maps to itself?'(10 votes)
A point is mapped to itself if it's in the same place before and after the transformation.(18 votes)
What does Sal mean by 'maps to itself?'(7 votes)- "map" is a term which shows the movement of a point, If you start with (3,2) and it moves along a vector <0,5>, we would say that (3,2) maps to (3,7) along the vector <0,5>. So if something maps to itself, it would (2,2) would map to itself (2.2) if it is reflected across y=x line. That also shows it is on the line of reflection. If you have a figure and rotate around one of the points, then that point would not change, thus mapping to itself while all the other points would change with the rotation.(11 votes)
Excuse me, however I do not understand this stuff at all, I guessed on the practice questions as well. Someone please help me, thanks.(10 votes)
What exactly is meant with 'the perpendicular bisector of segment PP'.
Any help would really be appreciated!(5 votes)- bisect means cut in half, perpendicular is right angles, so it is a line or line segment which is at right angles to and divides the segment in half.(8 votes)
- what is a perpendicular bisector and what does it mean for a point to map to itself(3 votes)
To understand what a perpendicular bisector is we need to understand both words separately.
When a line (or line segment/ray) is perpendicular to some other line it means that their intersection point forms four 90-degree (right) angles. As a visual example, a plus sign (+) is two perpendicular lines crossing each other and creating those four right angles.
A bisector in geometry can mean slightly different things depending on what we're talking about. In our case, we are talking about lines. When a given line A is a "bisector" of a given line B it means that line A intersects line B at the middle point (or midpoint) of line B. This time, the angle does not matter. We just care that the intersection point of the two lines is equally far apart (equidistant) from the endpoints of line segment B. In other words, line B is split in half by line A.
If we take those two terms together ("perpendicular" and "bisector") and combine them, we get two lines that intersect each other at right angles (this is perpendicularity). Additionally, we also know that one of these line segments splits the other one perfectly in half (bisection). Again, a plus (+) or multiplication (×) sign is a great example of two line segments that are perpendicular bisectors of each other.
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As for your other question, when a point maps to itself it simply means that the transformation applied to does not change the actual location of the point. In our case, the entire point of a reflection is to take a given point A some distance x from a line and mirror it on the opposite side of that line as a new point A' that is also the same distance x from it.
For example, if point A is 5cm away from our line of reflection, point A' will also be 5cm away from that same line. Obviously, for this to be an actual reflection, we have to remember that the line of reflection we are given will be the perpendicular bisector of the line segment formed by points A and A'.
If we extend this logic, we see why points that are on the line of reflection map to themselves. Their distance to the line is 0. Therefore, the distance of the reflected point to the line will also be zero. Thus, the location of the point remains unchanged, and we say that the point maps to itself.
Hope this helps!:)(12 votes)
what does "θ" mean(4 votes)- Theta is a greek letter used to represent unknown angles(9 votes)
- Isn't it possible for the last example to be a rotation?(4 votes)
Not with the information given to solve the problem.(6 votes)
3:00
Video transcript
- [Instructor] We're told
that a certain mapping in the xy-plane has the
following two properties. Each point on the line y is equal to three x minus two maps to itself. Any point P not on the line maps to a new point P' in such a way that the
perpendicular bisector of the segment PP' is the line y is equal to three x minus two. Which of the following statements is true? So is this describing a
reflection, a rotation, or a translation? So pause this video and see
if you can work through it on your own. All right so let me just
try to visualize this. So, and I'll just do a very quick, so if that's my y-axis, and that this right
over here is my x-axis. Three x minus two might
look something like this. The line three x minus two would look something like that. And so what we're saying is, or what they're telling us, is any point on this
after the transformation maps to itself. Now that by itself is a pretty good clue that we're likely dealing
with a reflection. Because remember with a reflection you reflect over a line, but if a point sits on the line, well it's just gonna
continue to sit on the line. But let's just make sure
that the second point is consistent with it being a reflection. So any point P not on the line, so let's see, point P, right over here, it maps to a new point P' in such a way that the perpendicular bisector of PP' is the line y equals three x minus two. So I need to connect, so the line three x minus two, y is equal to three x minus two, would be the perpendicular bisector of the segment between P and what? Well let's see I'd have to
draw a perpendicular line. It would have to have the
same length on both sides of the line y equals three x minus two. So P' would have to be right over there. So once again this is consistent
with being a reflection. P' is equidistant on the
other side of the line as P. So I definitely feel good that this is going to be a
reflection right over here. Let's do another example. So here we are told, and I'll switch my colors up, a certain mapping in the
plane has the following two properties. Point O maps to itself. Every point V on a circle C centered at O, all right, maps to a new point W on circle C so that the counterclockwise
angle from segment OV to OW measures 137 degrees. So is this a reflection,
rotation, or translation? Pause this video and try to
figure it out on your own. All right, so let's see. We're talking about circle centered at O. So let's see, let me just say, so I have this point O. It maps to itself on its transformation. Now every point V on
circle C centered at O. So let's see, let's say this is circle C centered at point O, so I'm gonna try to draw a
decent looking circle here. You get the idea. This is not the best hand-drawn
circle ever, all right. So every point, let's
just pick a point V here. So let's say that that is the point V, on a circle centered at
O maps to a new point W on the circle C. So maybe it maps to a new point W on, actually let me keep reading, W on circle C so that the
counterclockwise angle from OV to OW measures 137 degrees. Okay so we wanna know the angle from OV to OW going counterclockwise is 137 degrees. So this right over here is 137 degrees. And so this would be the segment OW. W would go right over there. And so what this looks like is well if we're talking about angles and we are rotating something, this point corresponds to this point, it's essentially the
point has been rotated by 137 degrees around point O. So this right over here
is clearly a rotation. This is a rotation. Sometimes reading this language at first is a little bit daunting. It was a little bit
daunting to me when I first (laughing) read it. But when you actually just break it down and you actually try to
visualize what's going on, you'll say well okay
look they're just taking point V and they're
rotating it by 137 degrees around point O. And so this would be a rotation. Let's do one more example. So here we are told, so they're talking about, again a certain mapping in the xy-plane. Each circle O with radius r and centered at x y is
mapped to a circle O' with radius r and centered at x plus 11 and then y minus seven. So once again pause this
video, what is this? Reflection, rotation, or translation? All right so you might be tempted, if they're talking about circles like we did in the last example, maybe they're talking about a rotation. But look, what they're really saying is, is that if I have a circle, let's say I have a circle right over here, centered right over here. After it, so this is x comma y, centered at x comma y. It's mapped to a new circle
O' with the same radius. So if this is the radius, it's mapped to a new circle
with the same radius, but now it is centered at, now it is centered at x plus 11, so our new x-coordinate
is gonna be 11 larger, x plus 11, and our y-coordinate
is gonna be seven less. But we have the exact same radius. We have the exact same radius. So our circle would still, so we have the exact same
radius right over here. So what just happened to the circle? Well we kept the radius the same, and we just shifted, we just shifted our center to the right by 11, plus 11, and we shifted it down by seven. We shifted it down by seven. So this is clearly a translation. So we would select that right over there. And we're done.