Question

Graph a triangle (XYZ) and reflect it over the line y=x to create triangle X'Y'Z'. Describe the transformation using words. Draw a line segment from point X to the reflecting line, and then draw a line segment from point X' to the reflecting line. What do you notice about the two line segments you drew? Do you think you would see the same characteristic if you drew the line segment connecting Y with the reflecting line and then Y' with the reflecting line? How do you know? (10 points)

Answer 1

let's break this down into steps:

1. graph a triangle XYZ
you can do this either by hand or with graphing software like desmos. pick any three coordinates (small integers are recommended) to be your X, Y, and Z points

2. reflect over the line y = x
start by drawing the line y = x (should be a straight line through the origin, with slope of positive 1)

then, treat this line as a mirror, and reflect each point X, Y, and Z across the line. I will demonstrate with the point X(0,5) that I made up.

Answer 2

now, to reflect the point, I consider what the "mirror image" of X would look like if I reflected it across y = x. remember, the two points must be symmetric about y = x.

I'll call the reflected image X'



now, if you're having trouble visualizing it, you can also use a formula to determine the reflection across y = x. simply swapping the x and y-coordinates of your point will generate the reflection.

apply this logic and transform your chosen X, Y, and Z values

Answer 3

3. Draw a line segment from point X to the reflecting line, and then draw a line segment from point X' to the reflecting line.

in this case, the reflecting line is y = x. so we simply draw the shortest possible line from X to y = x (I'll use blue) like so:

Answer 4

now, it also asks for a line from X' to y = x, so I will do that as well (I'll use orange):

Answer 5

4. What do you notice about the two line segments you drew?

Notice anything about the blue and orange segments? Pay attention to their lengths

5. Do you think you would see the same characteristic if you drew the line segment connecting Y with the reflecting line and then Y' with the reflecting line?

Any thoughts about what would happen if you did the same thing with Y and Y'? Would it be the same outcome as with X and X'? If you're having trouble, try repeating step 3 with Y and Y'

6. How do you know?
Think about the definition of a reflection - what must be true about the distances between the original point, the line of reflection, and the reflected image?

If you're having trouble, please review the mathematical definition of a reflection

https://www.mathsisfun.com/geometry/reflection.html

Answer 6

I know this is a lot, but please feel free to ask for help with any of the steps, or to check your work