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The coordinates of the vertices of 
/_\DEF are 
D(2,3),E(4,0), and 
F(1,-2). The coordinates of the vertices of 
/_\TVW are 
T(0,3), 
V(-2,0), and 
W(1,-2).
Which series of transformations correctly shows that 
/_\DEF~=/_\TVW?
a reflection over the 
y-axis and a translation 2 units right
a reflection over the 
x-axis and a translation 2 units down
a reflection over the 
x-axis and a translation 2 units up

The coordinates of the vertices of DEF \triangle D E F are D(2,3),E(4,0) D(2,3), E(4,0) , and F(1,2) F(1,-2) . The coordinates of the vertices of TVW \triangle T V W are T(0,3) T(0,3) , V(2,0) V(-2,0) , and W(1,2) W(1,-2) .\newlineWhich series of transformations correctly shows that DEFTVW? \triangle D E F \cong \triangle T V W ? \newlinea reflection over the y y -axis and a translation 22 units right\newlinea reflection over the x x -axis and a translation 22 units down\newlinea reflection over the x x -axis and a translation 22 units up

Full solution

Q. The coordinates of the vertices of DEF \triangle D E F are D(2,3),E(4,0) D(2,3), E(4,0) , and F(1,2) F(1,-2) . The coordinates of the vertices of TVW \triangle T V W are T(0,3) T(0,3) , V(2,0) V(-2,0) , and W(1,2) W(1,-2) .\newlineWhich series of transformations correctly shows that DEFTVW? \triangle D E F \cong \triangle T V W ? \newlinea reflection over the y y -axis and a translation 22 units right\newlinea reflection over the x x -axis and a translation 22 units down\newlinea reflection over the x x -axis and a translation 22 units up
  1. Reflect over y-axis: Step 11: Reflect /_DEFDEF over the y-axis.\newlineTo reflect a point (x,y)(x, y) over the y-axis, change the x-coordinate to x-x. So:\newlineD(2,3)(2, 3) becomes D'(2,3)(-2, 3),\newlineE(4,0)(4, 0) becomes E'(4,0)(-4, 0),\newlineF(1,2)(1, -2) becomes F'(1,2)(-1, -2).
  2. Translate 22 units right: Step 22: Translate DD', EE', FF' 22 units to the right.\newlineTo translate a point (x,y)(x, y) 22 units right, add 22 to the xx-coordinate. So:\newlineD(2,3)D'(-2, 3) becomes D(0,3)D''(0, 3),\newlineEE'00 becomes EE'11,\newlineEE'22 becomes EE'33.
  3. Compare coordinates after transformations: Step 33: Compare the coordinates of TVW\triangle TVW with DEF\triangle DEF after transformations.\newlineT(0,3)T(0, 3), V(2,0)V(-2, 0), and W(1,2)W(1, -2) are the coordinates of TVW\triangle TVW.\newlineD(0,3)D''(0, 3), E(2,0)E''(-2, 0), and F(1,2)F''(1, -2) are the coordinates of DEF\triangle DEF after transformations.\newlineThe coordinates match, indicating that DEF\triangle DEF is congruent to TVW\triangle TVW after the specified transformations.

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