The vertices of triangle ABC are A(2,2) B(5,2) and C(5,7). Find the vertices of the image if triangle ABC is
(A) translated 3 units to the left and 2 units downward
(B) reflected about the y-axis
(C) rotated 90 degrees about the origin in a clockwise directio
A (x,y) -> (x-3,y-2)
B (x,y) -> (-x,y)
C (x,y) -> (y,-x)
To find the vertices of the image of a triangle when it undergoes different transformations, we'll need to know the rules for each transformation.
(A) Translating a triangle involves shifting all of its vertices by a certain amount horizontally (left or right) and vertically (up or down). To translate triangle ABC 3 units to the left and 2 units downward, we simply subtract 3 from the x-coordinates and subtract 2 from the y-coordinates of each vertex:
A'(x,y) = A(2-3, 2-2) = A(-1, 0)
B'(x,y) = B(5-3, 2-2) = B(2, 0)
C'(x,y) = C(5-3, 7-2) = C(2, 5)
So, the vertices of the image of triangle ABC after translation are A'(-1, 0), B'(2, 0), and C'(2, 5).
(B) Reflecting a triangle about the y-axis means flipping it horizontally. To reflect triangle ABC about the y-axis, we need to negate the x-coordinates of each vertex while keeping the y-coordinates unchanged:
A'(x,y) = A(-x, y) = A(-2, 2)
B'(x,y) = B(-x, y) = B(-5, 2)
C'(x,y) = C(-x, y) = C(-5, 7)
Therefore, the vertices of the image of triangle ABC after reflecting about the y-axis are A'(-2, 2), B'(-5, 2), and C'(-5, 7).
(C) Rotating a triangle 90 degrees clockwise about the origin involves swapping the x and y coordinates and changing the sign of the new x-coordinate. To rotate triangle ABC 90 degrees clockwise, we obtain the following coordinates:
A'(x,y) = A(y, -x) = A(2, -2)
B'(x,y) = B(y, -x) = B(2, -5)
C'(x,y) = C(y, -x) = C(7, -5)
So, the vertices of the image of triangle ABC after rotating 90 degrees clockwise about the origin are A'(2, -2), B'(2, -5), and C'(7, -5).