Question

graph the image of △abc after the following sequence of transformations: rotation 180° counterclockwise around the origin reflection across the line x = -1

Explanation:

Step1: Apply rotation rule

The rule for a 180 - degree counter - clockwise rotation around the origin is $(x,y)\to(-x,-y)$. Let the coordinates of $A=(8, - 4)$, $B=(6,-7)$, $C=(8,-10)$. After rotation, $A'=(-8,4)$, $B'=(-6,7)$, $C'=(-8,10)$.

Step2: Apply reflection rule

The rule for reflecting a point $(x,y)$ across the line $x = a$ is $(x,y)\to(2a - x,y)$. Here $a=-1$. For point $A'(-8,4)$, the new $x$ - coordinate is $2\times(-1)-(-8)=-2 + 8=6$, so $A''=(6,4)$. For point $B'(-6,7)$, the new $x$ - coordinate is $2\times(-1)-(-6)=-2 + 6 = 4$, so $B''=(4,7)$. For point $C'(-8,10)$, the new $x$ - coordinate is $2\times(-1)-(-8)=-2 + 8=6$, so $C''=(6,10)$.

Step3: Graph the new triangle

Plot the points $A''(6,4)$, $B''(4,7)$, $C''(6,10)$ on the given coordinate grid and connect them to form the new triangle.

Answer:

Graph the triangle with vertices $A''(6,4)$, $B''(4,7)$, $C''(6,10)$ on the provided grid.