Right triangle ABC and its image, triangle A'B'C' are shown in the image attached. Algebraically prove that a clockwise and counterclockwise rotation of 180° about the origin for triangle ABC are equivalent rotations.
Accepted Solution
A:
Answer:See explanationStep-by-step explanation:Triangle ABC ha vertices at: A(-3,6), B(0,-4) and (2,6).Let us apply 90 degrees clockwise about the origin twice to obtain 180 degrees clockwise rotation.We apply the 90 degrees clockwise rotation rule.[tex](x,y)\to (y,-x)[/tex][tex]\implies A(-3,6)\to (6,3)[/tex][tex]\implies B(0,4)\to (4,0)[/tex][tex]\implies C(2,6)\to (6,-2)[/tex]We apply the 90 degrees clockwise rotation rule again on the resulting points:[tex]\implies (6,3)\to A''(3,-6)[/tex][tex]\implies (4,0)\to B''(0,-4)[/tex][tex]\implies (6,-2)\to C''(-2,-6)[/tex]Let us now apply 90 degrees counterclockwise rotation about the origin twice to obtain 180 degrees counterclockwise rotation.We apply the 90 degrees counterclockwise rotation rule.[tex](x,y)\to (-y,x)[/tex][tex]\implies A(-3,6)\to (-6,-3)[/tex][tex]\implies B(0,4)\to (-4,0)[/tex][tex]\implies C(2,6)\to (-6,2)[/tex]We apply the 90 degrees counterclockwise rotation rule again on the resulting points:[tex]\implies (-6,-3)\to A''(3,-6)[/tex][tex]\implies (-4,0)\to B''(0,-4)[/tex][tex]\implies (-6,2)\to C''(-2,-6)[/tex]We can see that A''(3,-6), B''(0,-4) and C''(-2,-6) is the same for both the 180 degrees clockwise and counterclockwise rotations.