Question

find a congruence transformation that maps $delta agk$ to $delta qct$

which congruence transformation maps the preimage to the congruent image? select the correct choice below and fill in the answer boxes to complete you choice

a. $t_{\square\square}\circ r_{x-\text{axis}}(\delta agk)$
b. $t_{\square\square}\circ r_{(270^{\circ},o)}(\delta agk)$
c. $t_{\square\square}\circ r_{y-\text{axis}}(\delta agk)$
d. $t_{\square\square}\circ r_{(90^{\circ},o)}(\delta agk)$

Explanation:

Step1: Identify preimage and image

Preimage: $\Delta AGK$ (lower grid), Image: $\Delta QCT$ (upper grid)

Step2: Analyze rotation first

Notice $\Delta AGK$ rotated 180° about origin maps to a shape aligned with $\Delta QCT$ but shifted. Rotation rule: $R_{O,180^\circ}(x,y)=(-x,-y)$

Step3: Calculate translation vector

Take point $A(2,-4)$: rotated 180° is $(-2,4)$. Image point $Q(4,8)$. Translation: $(4-(-2),8-4)=(6,4)$. So $T_{\langle 6,4
angle}$

Step4: Verify composition

Apply $T_{\langle 6,4
angle} \circ R_{O,180^\circ}$ to all points:

• $G(4,-6)$: rotated is $(-4,6)$, translated is $(-4+6,6+4)=(2,10)$ which matches $C(2,10)$
• $K(1,-3)$: rotated is $(-1,3)$, translated is $(-1+6,3+4)=(5,7)$ which matches $T(5,7)$

Answer:

A. $T_{\langle 6,4
angle} \circ R_{O,180^\circ}(\Delta AGK)$