Question

question 1-14
triangle abc with coordinates a(-3,0), b(-1,0), and c(-3,-4) was transformed to create triangle abc with coordinates a-6,2, b-4,2, and c-6,-2 (the rest of the coordinates for a, b, c might be partially obscured but the key part is the transformation of triangle abc to triangle abc. select all that apply.
options:
(x,y) → (x + 3, y - 2)
(x,y) → (x - 3, y + 2)
(x,y) → (x - 2, y + 3)
(x,y) → (x + 2, y - 3)
triangle abc translates 3 units to the left and 2 units up.
triangle abc translates 2 units to the right and 3 units down.

Explanation:

Step1: Analyze translation of point A

Original \( A(-3,0) \), new \( A'(-6,2) \). For \( x \)-coordinate: \( -3 + a=-6\Rightarrow a = - 3 \); for \( y \)-coordinate: \( 0 + b = 2\Rightarrow b = 2 \). So translation rule \( (x,y)\to(x - 3,y + 2) \).

Step2: Analyze translation of point B

Original \( B(-1,0) \), new \( B'(-4,2) \). \( x \): \( -1-3=-4 \), \( y \): \( 0 + 2 = 2 \). Matches \( (x,y)\to(x - 3,y + 2) \).

Step3: Analyze translation of point C

Original \( C(-3,-4) \), new \( C'(-6,-2) \). \( x \): \( -3-3=-6 \), \( y \): \( -4 + 2=-2 \). Matches \( (x,y)\to(x - 3,y + 2) \).

Step4: Interpret translation direction

\( x \)-coordinate decreases by 3 (left 3 units), \( y \)-coordinate increases by 2 (up 2 units). So "Triangle ABC translates 3 units to the left and 2 units up" is correct.

Step5: Check other options

• \( (x,y)\to(x + 3,y - 2) \): \( x \) increases, \( y \) decreases, wrong.
• \( (x,y)\to(x - 2,y + 3) \): \( x \) change \( -2

eq - 3 \), \( y \) change \( 3
eq2 \), wrong.

• \( (x,y)\to(x + 2,y - 3) \): \( x \) increases, \( y \) decreases, wrong.
• "Triangle ABC translates 2 units to the right and 3 units down": Opposite direction, wrong.

Answer:

\((x,y)\to(x - 3,y + 2)\), Triangle ABC translates 3 units to the left and 2 units up.