A 180^(∘) rotation about the origin changes point (x, y) into point (- x,- y).
P''(1,2), Q''(1,0), R''(-2,0)
Practice makes perfect
We know that our starting triangle has the following vertices.
P(-1, 2), Q(-1, 0), R(2,0)
We want to rotate this triangle by 180^(∘) about the origin and then we will reflect it in the x-axis. Our goal is to find the coordinates of the image after the two transformations. Let's take a look at each of these transformations one at a time, beginning with the rotation.
Rotation
When a triangle is rotated by 180^(∘) about the origin, the coordinates of the vertices of the triangle always change in the same way.
( x, y)→ ( - x, - y)
We can use this rule with the given points to find the coordinates of the rotated triangle.
(x, y)
(- x, - y)
Simplify
P( -1, 2)
P'( -( -1), - 2)
P'(1,-2)
Q( -1, 0)
Q'( -( -1), - 0)
Q'(1,0)
R( 2, 0)
R'( - 2, - 0)
R'(-2,0)
We could also draw the original triangle and then rotate it to get these coordinates.
Reflection
A reflection in the x-axis is a transformation where the x-coordinates stay the same but the y-coordinates change signs. The y-coordinates become the opposite numbers.
( x, y)
⟶
( x, - y)
Let's apply this rule to the vertices of the rotated triangle.
(x, y)
(x, - y)
Simplify
P'( 1, -2)
P''( 1, -( -2))
P''(1,2)
Q'( 1, 0)
Q''( 1, - 0)
Q''(1,0)
R'( -2, 0)
R''( -2, - 0)
R''(-2,0)
Now we know the coordinates of the triangle after the two transformations are finished.
P''(1,2), Q''(1,0), R''(-2, 0)
Once again, we could also get these points by doing the transformation on the graph. Let's see how the two transformations combined change the original triangle.
