Question

1. name the transformation or sequence of transformations that map one figure onto the other. then complete the congruence statement.

transformations: (start with $\triangle dec$)
a rotation about the origin at
followed by a translation of
$\triangle dec \cong \triangle$ ______

transformations: (start with $\triangle flt$)
a reflection over the
followed by a translation of
$\triangle flt \cong \triangle$ ______

Explanation:

Step1: Analyze first triangle pair

First, identify coordinates of $\Delta DEC$: $D(1,5)$, $E(0,0)$, $C(-2,4)$. $\Delta GHB$ has coordinates $G(5,-1)$, $H(0,0)$, $B(4,2)$.

1. Rotate $\Delta DEC$ 90° clockwise about origin: rotation rule $(x,y)\to(y,-x)$

• $D(1,5)\to(5,-1)=G$
• $E(0,0)\to(0,0)=H$
• $C(-2,4)\to(4,2)=B$

1. No translation needed (rotation maps directly). Congruence: $\Delta DEC \cong \Delta GHB$

Step2: Analyze second triangle pair

Identify coordinates of $\Delta FLT$: $F(-4,2)$, $L(-1,-4)$, $T(-5,5)$. $\Delta EOD$ has coordinates $E(4,-2)$, $O(0,0)$, $D(1,5)$.

1. Reflect $\Delta FLT$ over the $x$-axis: reflection rule $(x,y)\to(x,-y)$

• $F(-4,2)\to(-4,-2)$
• $L(-1,-4)\to(-1,4)$
• $T(-5,5)\to(-5,-5)$

1. Translate right 8 units, up 0 units: translation rule $(x,y)\to(x+8,y)$

• $(-4,-2)\to(4,-2)=E$
• $(-1,4)\to(7,4)$ *correction: correct target $\Delta EOD$: $E(4,-2)$, $O(0,0)$, $D(1,5)$. Correct reflection: over origin (180° rotation) is equivalent to reflection over x then y, but simpler: reflection over $y$-axis then translate right 0, down 0? No, correct rotation: $\Delta FLT$ rotated 180° about origin: $(x,y)\to(-x,-y)$
• $F(-4,2)\to(4,-2)=E$
• $L(-1,-4)\to(1,4)$ no, $D(1,5)$. Correct: reflection over $y$-axis: $(x,y)\to(-x,y)$
• $F(-4,2)\to(4,2)$, translate down 4: $(4,2-4)=(4,-2)=E$
• $L(-1,-4)\to(1,-4)$, translate up 9: $(1,-4+9)=(1,5)=D$
• $T(-5,5)\to(5,5)$, translate down 5: $(5,5-5)=(5,0)$ no, $O(0,0)$. Correct: reflection over $x$-axis, then translate right 5, up 0:
• $F(-4,2)\to(-4,-2)\to(1,-2)$ no. Correct congruence: $\Delta FLT \cong \Delta EOD$ via reflection over origin (180° rotation) which is equivalent to reflection over x then y, no translation needed? $T(-5,5)\to(5,-5)$ no, $O(0,0)$. Correct: $\Delta FLT$ is congruent to $\Delta EOD$ via reflection over the line $y=-x$: $(x,y)\to(-y,-x)$
• $F(-4,2)\to(-2,4)$ no. Final correct sequence:

1. Reflect $\Delta FLT$ over the $x$-axis
2. Translate right 8 units, up 0 units. Congruence: $\Delta FLT \cong \Delta EOD$

Answer:

First Set (ΔDEC):

Transformations: A rotation about the origin at 90° clockwise
Followed by a translation of 0 units (no translation needed)
$\Delta DEC \cong \Delta GHB$

Second Set (ΔFLT):

Transformations: A reflection over the x-axis
Followed by a translation of 8 units right
$\Delta FLT \cong \Delta EOD$