Graph these points.
Angle DFG has vertices D (–2, 4), F (–3, 1), and G (–1, 2). Find the vertices of the image after the rotation of 180° about the origin.
Rotation 180 degrees around (0,0) takes (x,y) --> (-x,-y)
Not much challenge here:
D:(-2,4) --> (2,-4)
. . .
*a triangle PQR is of coordinates P(1,3);Q(4,3)and R(4,-3)
a) plot the triangle PQR in X, Y- plane and join them to form a triangle.
b) Use a translation(2,-3)to find the image of PQR ; plot this image in the same plane as (a) above.
c) rotation the triangle PQR about origin with - 180;don't change the plane.
d) what is the image of triangle PQR under reflection of y=0
To find the vertices of the image after a rotation of 180° about the origin, you need to rotate each point individually.
The formula for rotating a point (x, y) by an angle of 180° about the origin is:
(x', y') = (-x, -y)
Let's apply this formula to the given points:
For point D (–2, 4):
x' = -(-2) = 2
y' = -(4) = -4
So the coordinates of the image of point D after the rotation are (2, -4).
For point F (–3, 1):
x' = -(-3) = 3
y' = -(1) = -1
So the coordinates of the image of point F after the rotation are (3, -1).
For point G (–1, 2):
x' = -(-1) = 1
y' = -(2) = -2
So the coordinates of the image of point G after the rotation are (1, -2).
Therefore, the vertices of the image after the rotation of 180° about the origin are:
D' (2, -4), F' (3, -1), and G' (1, -2).
To graph the points of angle DFG and find the image after a 180° rotation about the origin, follow these steps:
1. Plot the given points D (-2, 4), F (-3, 1), and G (-1, 2) on a coordinate plane.
2. With a straightedge, connect the points to form angle DFG.
3. To perform a rotation of 180° about the origin, each point's coordinates need to be multiplied by -1.
4. Apply the rotation to each point:
- Point D (x, y) = (-2, 4) after rotation becomes (-(-2), -4) = (2, -4).
- Point F (x, y) = (-3, 1) after rotation becomes (-(-3), -1) = (3, -1).
- Point G (x, y) = (-1, 2) after rotation becomes (-(-1), -2) = (1, -2).
5. Plot the new points after the rotation: D' (2, -4), F' (3, -1), and G' (1, -2).
6. Connect the new points to form the image of angle DFG after the 180° rotation about the origin.
This completes the process of graphing the original points and finding the image after the rotation.