Question

1. the coordinates of two points, and their images, are given below

a (-2, 4) a (-4, -2) b(6, -3) b(3, 6)
what transformation occurred to map points a and b to their images?

1. john draw the following shapes, and their images, on the plan below

a) what transformation did john preform on δabc to draw δabc?

b) what transformation did john preform on δabc to draw δabc?

1. rotate the coordinate (3, 5) 90° counter - clockwise around the point (1, -8)

1. rotate the coordinate (-1, 6) 270° clockwise around the point (4, -2)

Explanation:

Problem 18

Step1: Analyze Point A mapping

Compare $A(-2,4)$ to $A'(-4,-2)$.
Check scaling: $x$: $\frac{-4}{-2}=2$, $y$: $\frac{-2}{4}=-\frac{1}{2}$ → not uniform scaling.
Check rotation: For 90° clockwise rotation, $(x,y)\to(y,-x)$: $(-2,4)\to(4,2)$ ≠ $A'$. For 180° rotation, $(x,y)\to(-x,-y)$: $(-2,4)\to(2,-4)$ ≠ $A'$. For 90° counterclockwise, $(x,y)\to(-y,x)$: $(-2,4)\to(-4,-2)=A'$.

Step2: Verify with Point B

Test 90° counterclockwise rotation on $B(6,-3)$: $(x,y)\to(-y,x)$ → $-(-3)=3$, $6$ → $(3,6)=B'$.

A) Compare coordinates of $\triangle ABC$: $A(-3,5)$, $B(-5,1)$, $C(-3,3)$ and $\triangle A'B'C'$: $A'(-4,0)$, $B'(-6,-4)$, $C'(-2,-2)$. Calculate vertical/horizontal shifts: $A$ to $A'$: $x: -3-1=-4$, $y:5-5=0$; $B$ to $B'$: $x:-5-1=-6$, $y:1-5=-4$; $C$ to $C'$: $x:-3+1=-2$, $y:3-5=-2$. This is a translation.
B) Compare $\triangle A'B'C'$: $A'(-4,0)$, $B'(-6,-4)$, $C'(-2,-2)$ and $\triangle A''B''C''$: $A''(4,0)$, $B''(6,-4)$, $C''(2,-2)$. $x$-coordinates are negated, $y$-coordinates stay the same, which is reflection over the y-axis.

Step1: Translate point to origin

Subtract center $(1,-8)$ from $(3,5)$:
$(3-1, 5-(-8))=(2,13)$

Step2: 90° counterclockwise rotation

Rotation rule: $(x,y)\to(-y,x)$
$(2,13)\to(-13,2)$

Step3: Translate back to original center

Add center $(1,-8)$ to the result:
$(-13+1, 2+(-8))=(-12,-6)$

Answer:

A 90° counterclockwise rotation about the origin.

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Problem 19