Question

follow the guided instructions below to rotate the figure 90° clockwise about the point (3, 1). fill out all of the corresponding points of the image figure. pre - image image (2, - 5) → ( ) (5, - 4) → ( ) (6, - 1) → ( ) (2, - 3) → ( )

Explanation:

Step1: Translate points

First, translate the given points so that the center of rotation $(3,1)$ becomes the origin. For a point $(x,y)$ and center of rotation $(a,b)=(3,1)$, the translated point $(x',y')=(x - a,y - b)$.

Step2: Apply 90 - degree clock - wise rotation formula

The formula for a 90 - degree clock - wise rotation of a point $(x',y')$ about the origin is $(y',-x')$.

Step3: Translate back

After rotation, translate the points back to the original coordinate system. If the rotated - translated point is $(x_1',y_1')$, the final point $(x_1,y_1)=(x_1'+a,y_1'+b)$.

For the point $(2,-5)$:

• Translate: $x'=2 - 3=-1,y'=-5 - 1=-6$.
• Rotate: $x_1'=-6,y_1' = 1$.
• Translate back: $x=-6 + 3=-3,y=1 + 1=2$.

For the point $(5,-4)$:

• Translate: $x'=5 - 3=2,y'=-4 - 1=-5$.
• Rotate: $x_1'=-5,y_1' = 2$.
• Translate back: $x=-5+3=-2,y=2 + 1=3$.

For the point $(6,-1)$:

• Translate: $x'=6 - 3=3,y'=-1 - 1=-2$.
• Rotate: $x_1'=-2,y_1' = 3$.
• Translate back: $x=-2 + 3=1,y=3 + 1=4$.

For the point $(2,-3)$:

• Translate: $x'=2 - 3=-1,y'=-3 - 1=-4$.
• Rotate: $x_1'=-4,y_1' = 1$.
• Translate back: $x=-4 + 3=-1,y=1 + 1=2$.

Answer:

$(2,-5)\to(-3,2)$
$(5,-4)\to(-2,3)$
$(6,-1)\to(1,4)$
$(2,-3)\to(-1,2)$