Question

a point at (-3,5) is rotated 90 counterclockwise about the origin, then reflected across the line y = 3, and finally translated by the rule (x,y)→(x - 2,y + 1). what is the location of the final image?

• a) (-7,10)
• b) (-5,-3)
• c) (7,-2)
• d) (-7,8)

a) (-7,10)
b) (-5,-3)
c) (7,-2)
d) (-7,8)

Explanation:

Step1: Apply 90 - degree counter - clockwise rotation

The rule for a 90 - degree counter - clockwise rotation about the origin is $(x,y)\to(-y,x)$. For the point $(-3,5)$, we have $(-3,5)\to(-5,-3)$.

Step2: Reflect across the line $y = 3$

The distance between $y=-3$ and $y = 3$ is $d=3-(-3)=6$. The new $y$ - coordinate after reflection across $y = 3$ is $3+(3 - (-3))=9$, and the $x$ - coordinate remains the same. So the point becomes $(-5,9)$.

Step3: Apply the translation rule

The translation rule is $(x,y)\to(x - 2,y + 1)$. For the point $(-5,9)$, we calculate $x=-5-2=-7$ and $y=9 + 1=10$. The final point is $(-7,10)$.

Answer:

A. $(-7,10)$