select the correct answer from each drop - down menu.
triangle abc, with vertices a(3, 0), b(2, 4), and c(4, 2), undergoes a transformation to form triangle abc, with vertices a(3, 0), b(2, - 4), and c(4, - 2). the type of transformation that triangle abc undergoes is a reflection across the x - axis
if triangle abc undergoes a transformation to form triangle abc, with vertices a(- 3, 0), b(- 2, - 4), and c(- 4, - 2), then the type of transformation that triangle abc undergoes is a
drop - down menu options: 180° clockwise rotation, reflection across the x - axis, reflection across the y - axis, translation 1 unit right and 1 unit up, dilation by a scale factor of 0.5

Question

Accepted Answer

### First Transformation (ABC to A'B'C'):
# Explanation:
## Step1: Analyze coordinates of ABC and A'B'C'
- $ A(3, 0) \to A'(3, 0) $
- $ B(2, 4) \to B'(2, -4) $
- $ C(4, 2) \to C'(4, -2) $
## Step2: Identify transformation rule
The x - coordinates remain the same, and the y - coordinates are multiplied by - 1. This is the rule for reflection across the x - axis ($(x,y)\to(x, - y)$).

### Second Transformation (A'B'C' to A''B''C''):
# Explanation:
## Step1: Analyze coordinates of A'B'C' and A''B''C''
- $ A'(3, 0) \to A''(-3, 0) $
- $ B'(2, - 4) \to B''(-2, - 4) $
- $ C'(4, - 2) \to C''(-4, - 2) $
## Step2: Identify transformation rule
The y - coordinates remain the same, and the x - coordinates are multiplied by - 1. This is the rule for reflection across the y - axis ($(x,y)\to(-x,y)$). But wait, another way: For a $ 180^{\circ} $ rotation, the rule is $(x,y)\to(-x, - y)$. Wait, let's check: From ABC to A''B''C'': $ A(3,0)\to A''(-3,0) $ (x - multiplied by - 1, y same? No, wait ABC to A''B''C'': Wait, first ABC to A'B'C' is reflection over x - axis (y becomes - y), then A'B'C' to A''B''C'': $ A'(3,0)\to A''(-3,0) $ (x becomes - x, y same), $ B'(2,-4)\to B''(-2,-4) $ (x becomes - x, y same), $ C'(4,-2)\to C''(-4,-2) $ (x becomes - x, y same). Wait, no, wait the coordinates of A''B''C'' are $ A''(-3,0) $, $ B''(-2,-4) $, $ C''(-4,-2) $. From A'B'C' ($ A'(3,0) $, $ B'(2,-4) $, $ C'(4,-2) $) to A''B''C'': The x - coordinates are multiplied by - 1, y - coordinates remain. But also, if we consider the transformation from ABC to A''B''C'': $ A(3,0)\to A''(-3,0) $, $ B(2,4)\to B''(-2,-4) $, $ C(4,2)\to C''(-4,-2) $. The rule here is $(x,y)\to(-x, - y)$, which is a $ 180^{\circ} $ clockwise (or counter - clockwise, since $ 180^{\circ} $ rotation is the same in both directions) rotation. But in the drop - down menu, for the second transformation (A'B'C' to A''B''C''), let's re - evaluate. Wait, A'B'C' has points $ A'(3,0) $, $ B'(2,-4) $, $ C'(4,-2) $. A''B''C'' has $ A''(-3,0) $, $ B''(-2,-4) $, $ C''(-4,-2) $. So the transformation from A'B'C' to A''B''C'' is $(x,y)\to(-x,y)$? No, $ B'(2,-4)\to B''(-2,-4) $: x becomes - x, y remains. $ A'(3,0)\to A''(-3,0) $: x becomes - x, y remains. $ C'(4,-2)\to C''(-4,-2) $: x becomes - x, y remains. Wait, but that's reflection over y - axis. But wait, the first transformation was reflection over x - axis, then reflection over y - axis is equivalent to a $ 180^{\circ} $ rotation. But in the drop - down menu, the options for the second transformation (after A'B'C'): Let's check the options again. The options are: $ 180^{\circ} $ clockwise rotation, reflection across x - axis, reflection across y - axis, translation 1 right 1 up, dilation 0.5.

Wait, let's re - express the coordinates:

From A'B'C' ($ (3,0) $, $ (2,-4) $, $ (4,-2) $) to A''B''C'' ($ (-3,0) $, $ (-2,-4) $, $ (-4,-2) $):

The transformation rule is $ (x,y)\to(-x,y) $? No, $ y $ is same, $ x $ is - x. So it's reflection across the y - axis? Wait no, reflection across y - axis is $ (x,y)\to(-x,y) $, which matches. But wait, another way: If we do a $ 180^{\circ} $ rotation on A'B'C', the rule is $ (x,y)\to(-x,-y) $. Let's check: $ A'(3,0)\to(-3,0) $ (which is A''), $ B'(2,-4)\to(-2,4) $, but B'' is $ (-2,-4) $. So that's not $ 180^{\circ} $ rotation. Wait, I made a mistake earlier. The first transformation: ABC to A'B'C' is reflection over x - axis (correct, since y becomes - y). Then A'B'C' to A''B''C'': $ A'(3,0)\to A''(-3,0) $ (x becomes - x, y same), $ B'(2,-4)\to B''(-2,-4) $ (x becomes - x, y same), $ C'(4,-2)\to C''(-4,-2) $ (x becomes - x, y same). So this is reflection over the y - axis? Wait, no, reflection over y - axis is $ (x,y)\to(-x,y) $, but here y is same. Wait, the y - coordinates in A'B'C' are $ 0, - 4, - 2 $, and in A''B''C'' they are $ 0, - 4, - 2 $. So yes, $ (x,y)\to(-x,y) $ when y is negative? Wait, no, the rule for reflection over y - axis is always $ (x,y)\to(-x,y) $, regardless of the sign of y. So in this case, since the y - coordinates are already negative (from the first reflection), reflecting over y - axis keeps the y - coordinate the same (because $ y$ in A'B'C' is $ y'=-y $ from original, then reflecting over y - axis: $ (x,y')\to(-x,y') $, which is what we see.

But let's go back to the problem's drop - down options for the first transformation (ABC to A'B'C'): The options include "reflection across the x - axis", which matches our first analysis.

For the second transformation (A'B'C' to A''B''C''): The options include "180° clockwise rotation"? Wait, no, let's recalculate the $ 180^{\circ} $ rotation rule: $ (x,y)\to(-x,-y) $. Let's apply that to A'B'C':

- $ A'(3,0)\to(-3,0) $ (matches A'')
- $ B'(2,-4)\to(-2,4) $ (does not match B''(-2,-4))
- $ C'(4,-2)\to(-4,2) $ (does not match C''(-4,-2))

So $ 180^{\circ} $ rotation is not correct. Wait, the first transformation is reflection over x - axis (ABC to A'B'C'), then the second transformation: A'B'C' to A''B''C'': The x - coordinates are negated, y - coordinates same. So it's reflection over y - axis? But the drop - down option for the second transformation (after the first one) – wait, maybe I misread the problem. Let's re - examine the problem statement:

"The type of transformation that triangle ABC undergoes is a [first drop - down] (reflection across the x - axis is an option here, which we determined). If triangle A'B'C' undergoes a transformation to form triangle A''B''C'', with vertices $ A''(-3,0) $, $ B''(-2,-4) $, and $ C''(-4,-2) $, then the type of transformation that triangle A'B'C' undergoes is a [second drop - down]."

Wait, A'B'C' has vertices $ A'(3,0) $, $ B'(2,-4) $, $ C'(4,-2) $. A''B''C'' has $ A''(-3,0) $, $ B''(-2,-4) $, $ C''(-4,-2) $. So the transformation from A'B'C' to A''B''C'' is $ (x,y)\to(-x,y) $? No, $ y $ is same, $ x $ is - x. Wait, no, $ B'(2,-4)\to B''(-2,-4) $: $ x $ becomes - x, $ y $ remains. So this is reflection across the y - axis? But the drop - down options include "180° clockwise rotation", "reflection across x - axis", "reflection across y - axis", etc. Wait, maybe there's a mistake in my earlier analysis. Let's consider the composition of transformations. First, reflection over x - axis (ABC to A'B'C'), then reflection over y - axis (A'B'C' to A''B''C'') is equivalent to a $ 180^{\circ} $ rotation (since reflection over x - axis then y - axis is $ (x,y)\to(-x,-y) $? Wait no, reflection over x - axis: $ (x,y)\to(x,-y) $, then reflection over y - axis: $ (x,-y)\to(-x,-y) $, which is $ 180^{\circ} $ rotation. Wait, let's check with the coordinates:

- $ A(3,0)\to A'(3,0) $ (reflection over x - axis, y = 0, so no change), then $ A'(3,0)\to A''(-3,0) $ (reflection over y - axis: $ (3,0)\to(-3,0) $), which is $ (-x,-y) $ since $ y = 0 $, so $ (-3,0) $.
- $ B(2,4)\to B'(2,-4) $ (reflection over x - axis), then $ B'(2,-4)\to B''(-2,-4) $ (reflection over y - axis), which is $ (-2,-4)=(-x,-y) $ (since $ x = 2 $, $ y = 4 $, so $ -x=-2 $, $ -y=-4 $).
- $ C(4,2)\to C'(4,-2) $ (reflection over x - axis), then $ C'(4,-2)\to C''(-4,-2) $ (reflection over y - axis), which is $ (-4,-2)=(-x,-y) $ (since $ x = 4 $, $ y = 2 $, so $ -x=-4 $, $ -y=-2 $).

Ah! So the composition of reflection over x - axis and then reflection over y - axis is a $ 180^{\circ} $ rotation (because $ (x,y)\to(x,-y)\to(-x,-y) $, which is the $ 180^{\circ} $ rotation rule). So when going from A'B'C' to A''B''C'', since A'B'C' is $ (x,-y) $ from ABC, then applying $ 180^{\circ} $ rotation to A'B'C' (which is $ (x,-y) $): $ (x,-y)\to(-x,y) $? No, wait $ 180^{\circ} $ rotation is $ (x,y)\to(-x,-y) $, so for A'B'C' with coordinates $ (x,-y) $, $ 180^{\circ} $ rotation would be $ (x,-y)\to(-x,y) $, which does not match A''B''C''. Wait, I'm confused. Let's use the coordinates of A''B''C'' and see what transformation from A'B'C' gives that.

A'B'C' vertices: $ (3,0) $, $ (2,-4) $, $ (4,-2) $

A''B''C'' vertices: $ (-3,0) $, $ (-2,-4) $, $ (-4,-2) $

The change from A'B'C' to A''B''C'' is:

- $ x $: 3→ - 3, 2→ - 2, 4→ - 4 (multiply by - 1)
- $ y $: 0→0, - 4→ - 4, - 2→ - 2 (no change)

So the transformation is $ (x,y)\to(-x,y) $, which is reflection across the y - axis.

# Answer:
- The type of transformation that triangle ABC undergoes to form triangle A'B'C' is a **reflection across the x - axis**.
- The type of transformation that triangle A'B'C' undergoes to form triangle A''B''C'' is a **reflection across the y - axis** (or considering the composition, but based on the coordinates, reflection across y - axis). However, if we consider the composition of reflection over x - axis and then y - axis as a $ 180^{\circ} $ rotation, but based on the coordinate change from A'B'C' to A''B''C'', it's reflection across y - axis. But let's check the drop - down options again. The first drop - down: "reflection across the x - axis" is correct for ABC to A'B'C'. The second drop - down: From A'B'C' to A''B''C'', the x - coordinates are negated, y same, so "reflection across the y - axis" or "180° clockwise rotation"? Wait, no, $ 180^{\circ} $ rotation would negate both x and y, but here y is same. So the correct answer for the first transformation is "reflection across the x - axis", and for the second transformation (A'B'C' to A''B''C''), it's "reflection across the y - axis" or "180° clockwise rotation"? Wait, no, let's calculate the $ 180^{\circ} $ rotation on A'B'C':

$ 180^{\circ} $ rotation rule: $ (x,y)\to(-x,-y) $

- $ A'(3,0)\to(-3,0) $ (matches A'')
- $ B'(2,-4)\to(-2,4) $ (does not match B''(-2,-4))
- $ C'(4,-2)\to(-4,2) $ (does not match C''(-4,-2))

So $ 180^{\circ} $ rotation is incorrect. Therefore, the second transformation is reflection across the y - axis.

So the answers are:

First drop - down: reflection across the x - axis

Second drop - down: reflection across the y - axis

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QUESTION IMAGE

Question

Explanation:

First Transformation (ABC to A'B'C'):

Step1: Analyze coordinates of ABC and A'B'C'

Step2: Identify transformation rule

Second Transformation (A'B'C' to A''B''C''):

Step1: Analyze coordinates of A'B'C' and A''B''C''

Step2: Identify transformation rule

Answer: