After a triangle is rotated 180 degrees, its vertices are at (-4,1),(-1,4), and (-5,-8). What were the coordinates of the vertices before the rotation?
I think the answer is (1,-4), (4,-1),(8,-5). In rotation, the coordinates get switched. All I just did was switch the numbers and signs. Is this right?
do you want your rotation about the origin?
If so, then the line joining a point and its image point should pass through the origin, yours don't.
if you interchange the x and y coordinates you are forming a reflection in the line y=x
try switching the signs only and plotting the points.
You should have tried plotting right away.
To find the original coordinates of the vertices before the rotation, you need to understand how rotation affects the coordinates of points.
In a 2D rotation, each point (x, y) is transformed into a new point (x', y') by rotating it a specified number of degrees or radians around a fixed point, also known as the center of rotation.
In this case, the triangle has been rotated 180 degrees. To find the original coordinates, you can apply the reverse rotation by rotating the current coordinates 180 degrees. The formula to rotate a point (x, y) by a given angle θ around the origin (0, 0) is:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Since the triangle has been rotated 180 degrees, θ = 180° = π radians. Substituting this in the formula, we get:
x' = x * cos(π) - y * sin(π)
y' = x * sin(π) + y * cos(π)
Now, let's calculate the original coordinates using these formulas.
For the first vertex (-4, 1):
x' = (-4) * cos(π) - 1 * sin(π)
= -4 * (-1) - 1 * 0
= 4 - 0
= 4
y' = (-4) * sin(π) + 1 * cos(π)
= -4 * 0 + 1 * (-1)
= 0 - 1
= -1
So, the original coordinates for the first vertex are (4, -1).
By applying the same process to the other two vertices, you can find the original coordinates for the second and third vertices before the rotation.
For the second vertex (-1, 4):
x' = (-1) * cos(π) - 4 * sin(π)
= -1 * (-1) - 4 * 0
= 1 - 0
= 1
y' = (-1) * sin(π) + 4 * cos(π)
= -1 * 0 + 4 * (-1)
= 0 - 4
= -4
So, the original coordinates for the second vertex are (1, -4).
For the third vertex (-5, -8):
x' = (-5) * cos(π) - (-8) * sin(π)
= -5 * (-1) - (-8) * 0
= 5 - 0
= 5
y' = (-5) * sin(π) + (-8) * cos(π)
= -5 * 0 + (-8) * (-1)
= 0 - (-8)
= 8
So, the original coordinates for the third vertex are (5, 8).
Therefore, the original coordinates of the vertices before the rotation are (4, -1), (1, -4), and (5, 8).