Question

rectangle abcd is translated (x + 2, y - 3) and then rotated 180° about the origin. complete the table to show the locations of a, b, c, and d after both transformations. a (-5,1) a? b (-5,3) b? c (-1,3) c? d (-1,1) d? a(-2,-3), b(0,-3), c(0,1), d(-2,1) a(-3,-2), b(-3,0), c(1,0), d(1,-2) a(0,5), b(3,2), c(-1,2), d(-1,0) a(3,2), b(3,0), c(-1,0), d(-1,2)

Explanation:

Step1: Apply translation

For a point $(x,y)$ translated by $(x + 2,y - 3)$, we add 2 to the x - coordinate and subtract 3 from the y - coordinate.
For point A(-5,1): $A'=(-5 + 2,1-3)=(-3,-2)$
For point B(-5,3): $B'=(-5 + 2,3 - 3)=(-3,0)$
For point C(-1,3): $C'=(-1+2,3 - 3)=(1,0)$
For point D(-1,1): $D'=(-1 + 2,1-3)=(1,-2)$

Step2: Apply 180 - degree rotation

The rule for a 180 - degree rotation about the origin is $(x,y)\to(-x,-y)$.
For $A'(-3,-2)$: $A''=-(-3),-(-2)=(3,2)$
For $B'(-3,0)$: $B''=-(-3),-0=(3,0)$
For $C'(1,0)$: $C''=-1,-0=(-1,0)$
For $D'(1,-2)$: $D''=-1,-(-2)=(-1,2)$

Answer:

$A''(3,2), B''(3,0), C''(-1,0), D''(-1,2)$