How do I sketch the reflection of line segment AB about the line y = x?

  • MHB
  • Thread starter mathdad
  • Start date
  • Tags
    Reflection
In summary, the conversation discussed the process of sketching the reflection of a line segment AB about the line y = x. It was clarified that the line y = x is not the perpendicular bisector of AB, but rather the perpendicular bisector of the line segments from A to A' and B to B'. The conversation also mentioned finding two new points, A' and B', and determining the slope of the line segment AB.
  • #1
mathdad
1,283
1
The endpoints of line AB are given to be (-1, -2) and
(-5, -2). Sketch the reflection of line segment AB about the line y = x.

As I understand this concept, I must show that the line y = x is the perpendicular bisector of the line segment AB.

Two points are given. This tells me to find the slope m.

m = (-2 - (-2))/(-5 - (-1))

m = (-2 + 2)/(-5 + 1)

m = 0/-4

m = 0

When m = 0, we are talking about a horizontal line.

Where do I go from here?
 
Mathematics news on Phys.org
  • #2
A reflection about the line $y=x$ simply requires swapping the $x$ and $y$ coordinates.

To see why this is true, consider a point in the plane $\left(x_1,y_1\right)$. Its distance $d$ to the line $y=x$ is given by:

\(\displaystyle d=\frac{\left|x_1-y_1\right|}{\sqrt{2}}\)

So, we want to find the other point $\left(x_1,y_1\right)$ with the same distance, only on the other side of $y=x$, that lies along the line:

\(\displaystyle y=-(x-x_1)+y_1\)

Hence:

\(\displaystyle y_2-y_1=x_1-x_2\tag{1}\)

\(\displaystyle d=\frac{\left|x_2-y_2\right|}{\sqrt{2}}\)

This gives us two cases:

i) \(\displaystyle x_2-y_2=x_1-y_1\implies y_2-y_1=x_2-x_1\)

Adding this to (1), we find:

\(\displaystyle y_1=y_2\)

This means we have the original point, so we discard this.

ii) \(\displaystyle x_2-y_2=y_1-x_1\implies y_2+y_1=x_2+x_1\)

Adding this to (1), we find:

\(\displaystyle y_2=x_1\implies x_2=y_1\)

Thus, we find the reflection of $\left(x_1,y_1\right)$ about the line $y=x$ is the point $\left(y_1,x_1\right)$.
 
  • #3
MarkFL said:
A reflection about the line $y=x$ simply requires swapping the $x$ and $y$ coordinates.

To see why this is true, consider a point in the plane $\left(x_1,y_1\right)$. Its distance $d$ to the line $y=x$ is given by:

\(\displaystyle d=\frac{\left|x_1-y_1\right|}{\sqrt{2}}\)

So, we want to find the other point $\left(x_1,y_1\right)$ with the same distance, only on the other side of $y=x$, that lies along the line:

\(\displaystyle y=-(x-x_1)+y_1\)

Hence:

\(\displaystyle y_2-y_1=x_1-x_2\tag{1}\)

\(\displaystyle d=\frac{\left|x_2-y_2\right|}{\sqrt{2}}\)

This gives us two cases:

i) \(\displaystyle x_2-y_2=x_1-y_1\implies y_2-y_1=x_2-x_1\)

Adding this to (1), we find:

\(\displaystyle y_1=y_2\)

This means we have the original point, so we discard this.

ii) \(\displaystyle x_2-y_2=y_1-x_1\implies y_2+y_1=x_2+x_1\)

Adding this to (1), we find:

\(\displaystyle y_2=x_1\implies x_2=y_1\)

Thus, we find the reflection of $\left(x_1,y_1\right)$ about the line $y=x$ is the point $\left(y_1,x_1\right)$.
Do I switch the original coodinates of the given points, plot the points on the xy-plane and then connect them with a straight line?
 
  • #4
RTCNTC said:
Do I switch the original coodinates of the given points, plot the points on the xy-plane and then connect them with a straight line?

Yes. (Yes)
 
  • #5
RTCNTC said:
The endpoints of line AB are given to be (-1, -2) and
(-5, -2). Sketch the reflection of line segment AB about the line y = x.

As I understand this concept, I must show that the line y = x is the perpendicular bisector of the line segment AB.
Well, there's your problem- you understand wrong. You need to find two new points, A' and B', such y= x is the perpendicular bisector of both the line segment from A to A' and the line segment from B to B'.

If A were (x, y) and B were (y, x), that is, if A were something like (2, 3) and B were (3, 2), then y= x would be the perpendicular bisector of AB.

Two points are given. This tells me to find the slope m.

m = (-2 - (-2))/(-5 - (-1))

m = (-2 + 2)/(-5 + 1)

m = 0/-4

m = 0

When m = 0, we are talking about a horizontal line.

Where do I go from here?
 
Last edited by a moderator:
  • #6
HallsofIvy said:
Well, there's your problem- you understand wrong. You need to find two new points, A' and B', such y= x is the perpendicular bisector of both the line segment from A to A' and the line segment from B to B'.

I know what to do thanks to Mark.
 

FAQ: How do I sketch the reflection of line segment AB about the line y = x?

1. What is the concept of "Reflection About y = x"?

The concept of "Reflection About y = x" involves reflecting or flipping a figure or shape over the line y = x. This means that any point (x,y) on the original figure will be reflected to the point (y,x) on the new figure. The line y = x is also known as the line of symmetry, as any point on one side of this line will have a corresponding point on the other side.

2. How is "Reflection About y = x" different from other types of reflections?

"Reflection About y = x" is different from other types of reflections because it involves reflecting over a specific line, y = x. Other types of reflections may involve reflecting over a different line or axis, such as the x-axis or y-axis. Additionally, "Reflection About y = x" will result in a figure that is rotated 90 degrees counterclockwise from the original figure.

3. What is the equation for "Reflection About y = x"?

The equation for "Reflection About y = x" is (x,y) -> (y,x). This means that for any point (x,y) on the original figure, the corresponding reflected point will be (y,x) on the new figure. This equation can also be written as (x,y) -> (x+y,y+x) to show the relationship between the original point and the reflected point.

4. How can "Reflection About y = x" be applied in real life?

"Reflection About y = x" can be applied in various ways in real life. For example, it can be used in architecture and design to create symmetrical structures or patterns. It is also commonly used in mathematics and physics to study the properties of reflection and symmetry. Additionally, it can be used in computer graphics to create reflections in 3D environments.

5. What other mathematical concepts are related to "Reflection About y = x"?

There are several other mathematical concepts that are related to "Reflection About y = x". One such concept is rotation, as "Reflection About y = x" can be thought of as a 90 degree rotation counterclockwise. Another related concept is symmetry, as the line y = x is a line of symmetry for the reflected figure. Additionally, the concepts of transformations and matrices are also related to "Reflection About y = x".

Similar threads

Replies
12
Views
2K
Replies
4
Views
976
Replies
1
Views
1K
Replies
3
Views
1K
Replies
6
Views
1K
Replies
3
Views
2K
Back
Top