name
geometry
8. on the graph below, draw and label $ \triangle p q r $, whose vertices are $ p(3,5), q(9,5) $, and $ r(7,7) $. on the same set of
axes, graph and state the coordinates of: $ \triangle p^{prime} q^{prime} r^{prime} $, the image of $ \triangle p q r $ after $ r_{90} $; $ \triangle p^{\prime \prime} q^{\prime \prime} r^{\prime \prime} $, the image of
$ \triangle p^{prime} q^{prime} r^{prime} $ after $ r_{x-\text { ans }} $; and $ \triangle p^{\prime \prime \prime} q^{\prime \prime \prime} r^{\prime \prime \prime} $, the image of $ \triangle p^{\prime \prime} q^{\prime \prime} r^{\prime \prime} $ after $ r_{y-\text { and }} $. based upon these graphs,
write a single transformation that shows the composition $ r_{y-\text { and }} \circ r_{x-\text { ans }} \circ r_{90} $.

9. given triangle $ a b c $ with coordinates $ a(-1,-2), b(0,-4) $, and $ c(3,-1) $. on the graph below, draw and label
$ \triangle a b c $. graph and label $ \triangle a^{\prime} b^{\prime} c^{\prime} $, the image of $ \triangle a b c $ after translation $ t_{4,-3} $. graph and label $ \triangle a^{\prime \prime} b^{\prime \prime} c^{\prime \prime} $, the
image of $ \triangle a^{\prime} b^{\prime} c^{\prime} $ after a reflection in the origin. graph and label $ \triangle a^{\prime \prime \prime} b^{\prime \prime \prime} c^{\prime \prime \prime} $, the image of $ \triangle a^{\prime \prime} b^{\prime \prime} c^{\prime \prime} $ after a
reflection in the line $ y=-x $.

Question

name
geometry
8. on the graph below, draw and label $ \triangle p q r $, whose vertices are $ p(3,5), q(9,5) $, and $ r(7,7) $. on the same set of
axes, graph and state the coordinates of: $ \triangle p^{prime} q^{prime} r^{prime} $, the image of $ \triangle p q r $ after $ r_{90} $; $ \triangle p^{\prime \prime} q^{\prime \prime} r^{\prime \prime} $, the image of
$ \triangle p^{prime} q^{prime} r^{prime} $ after $ r_{x-\text { ans }} $; and $ \triangle p^{\prime \prime \prime} q^{\prime \prime \prime} r^{\prime \prime \prime} $, the image of $ \triangle p^{\prime \prime} q^{\prime \prime} r^{\prime \prime} $ after $ r_{y-\text { and }} $. based upon these graphs,
write a single transformation that shows the composition $ r_{y-\text { and }} \circ r_{x-\text { ans }} \circ r_{90} $.

9. given triangle $ a b c $ with coordinates $ a(-1,-2), b(0,-4) $, and $ c(3,-1) $. on the graph below, draw and label
$ \triangle a b c $. graph and label $ \triangle a^{\prime} b^{\prime} c^{\prime} $, the image of $ \triangle a b c $ after translation $ t_{4,-3} $. graph and label $ \triangle a^{\prime \prime} b^{\prime \prime} c^{\prime \prime} $, the
image of $ \triangle a^{\prime} b^{\prime} c^{\prime} $ after a reflection in the origin. graph and label $ \triangle a^{\prime \prime \prime} b^{\prime \prime \prime} c^{\prime \prime \prime} $, the image of $ \triangle a^{\prime \prime} b^{\prime \prime} c^{\prime \prime} $ after a
reflection in the line $ y=-x $.

Accepted Answer

### Problem 8 Solution:

# Explanation:
## Step 1: Find $ \triangle P'Q'R' $ ( $ R_{90^\circ} $ rotation of $ \triangle PQR $)
The rule for a $ 90^\circ $ counterclockwise rotation about the origin is $ (x,y) \to (-y,x) $.
- For $ P(3,5) $: $ (-5,3) $ so $ P'(-5,3) $
- For $ Q(9,5) $: $ (-5,9) $ so $ Q'(-5,9) $
- For $ R(7,7) $: $ (-7,7) $ so $ R'(-7,7) $

## Step 2: Find $ \triangle P''Q''R'' $ ( $ r_{x-\text{axis}} $ reflection of $ \triangle P'Q'R' $)
The rule for reflection over the $ x $-axis is $ (x,y) \to (x,-y) $.
- For $ P'(-5,3) $: $ (-5,-3) $ so $ P''(-5,-3) $
- For $ Q'(-5,9) $: $ (-5,-9) $ so $ Q''(-5,-9) $
- For $ R'(-7,7) $: $ (-7,-7) $ so $ R''(-7,-7) $

## Step 3: Find $ \triangle P'''Q'''R''' $ ( $ r_{y-\text{axis}} $ reflection of $ \triangle P''Q''R'' $)
The rule for reflection over the $ y $-axis is $ (x,y) \to (-x,y) $.
- For $ P''(-5,-3) $: $ (5,-3) $ so $ P'''(5,-3) $
- For $ Q''(-5,-9) $: $ (5,-9) $ so $ Q'''(5,-9) $
- For $ R''(-7,-7) $: $ (7,-7) $ so $ R'''(7,-7) $

## Step 4: Determine the single transformation
Now, let's see the composition $ r_{y-\text{axis}} \circ r_{x-\text{axis}} \circ R_{90^\circ} $. Let's test the original point $ P(3,5) $:
- After $ R_{90^\circ} $: $ (-5,3) $ (as in Step 1)
- After $ r_{x-\text{axis}} $: $ (-5,-3) $ (as in Step 2)
- After $ r_{y-\text{axis}} $: $ (5,-3) $ (as in Step 3)

Now, let's check the rotation $ 90^\circ $ clockwise (or $ 270^\circ $ counterclockwise) or reflection. Wait, let's recall the rule for rotation $ 90^\circ $ clockwise: $ (x,y) \to (y,-x) $. Wait, no, let's re - evaluate.

Wait, the composition of $ r_{y - \text{axis}}\circ r_{x - \text{axis}}\circ R_{90^\circ} $. Let's use the general point $ (x,y) $:

1. First, $ R_{90^\circ} $: $ (x,y)\to(-y,x) $
2. Then, $ r_{x - \text{axis}} $: $ (-y,x)\to(-y,-x) $
3. Then, $ r_{y - \text{axis}} $: $ (-y,-x)\to(y,-x) $

The rule $ (x,y)\to(y,-x) $ is a $ 90^\circ $ clockwise rotation about the origin.

Let's verify with $ P(3,5) $: $ (3,5)\to(5,-3) $, which matches our earlier result. And for $ Q(9,5) $:

- $ R_{90^\circ} $: $ (-5,9) $
- $ r_{x - \text{axis}} $: $ (-5,-9) $
- $ r_{y - \text{axis}} $: $ (5,-9) $
Using the $ 90^\circ $ clockwise rule $ (x,y)\to(y,-x) $, for $ Q(9,5) $, we get $ (5,-9) $, which matches.

For $ R(7,7) $:

- $ R_{90^\circ} $: $ (-7,7) $
- $ r_{x - \text{axis}} $: $ (-7,-7) $
- $ r_{y - \text{axis}} $: $ (7,-7) $
Using the $ 90^\circ $ clockwise rule $ (x,y)\to(y,-x) $, for $ R(7,7) $, we get $ (7,-7) $, which matches.

So the composition $ r_{y-\text{axis}} \circ r_{x-\text{axis}} \circ R_{90^\circ} $ is equivalent to a $ 90^\circ $ clockwise rotation about the origin (or $ 270^\circ $ counterclockwise rotation about the origin).

### Problem 9 Solution:

# Explanation:
## Step 1: Find $ \triangle A'B'C' $ (Translation $ T_{4,-3} $ of $ \triangle ABC $)
The rule for translation $ T_{h,k} $ is $ (x,y)\to(x + h,y + k) $, where $ h = 4 $ and $ k=-3 $.
- For $ A(-1,-2) $: $ (-1 + 4,-2-3)=(3,-5) $ so $ A'(3,-5) $
- For $ B(0,-4) $: $ (0 + 4,-4-3)=(4,-7) $ so $ B'(4,-7) $
- For $ C(3,-1) $: $ (3 + 4,-1-3)=(7,-4) $ so $ C'(7,-4) $

## Step 2: Find $ \triangle A''B''C'' $ (Reflection in the origin of $ \triangle A'B'C' $)
The rule for reflection in the origin is $ (x,y)\to(-x,-y) $.
- For $ A'(3,-5) $: $ (-3,5) $ so $ A''(-3,5) $
- For $ B'(4,-7) $: $ (-4,7) $ so $ B''(-4,7) $
- For $ C'(7,-4) $: $ (-7,4) $ so $ C''(-7,4) $

## Step 3: Find $ \triangle A'''B'''C''' $ (Reflection in the line $ y=-x $ of $ \triangle A''B''C'' $)
The rule for reflection in the line $ y = -x $ is $ (x,y)\to(-y,-x) $.
- For $ A''(-3,5) $: $ (-5,3) $ so $ A'''(-5,3) $
- For $ B''(-4,7) $: $ (-7,4) $ so $ B'''(-7,4) $
- For $ C''(-7,4) $: $ (-4,7) $ so $ C'''(-4,7) $

# Answers:

## Problem 8:
- $ \triangle P'Q'R' $ coordinates: $ P'(-5,3) $, $ Q'(-5,9) $, $ R'(-7,7) $
- $ \triangle P''Q''R'' $ coordinates: $ P''(-5,-3) $, $ Q''(-5,-9) $, $ R''(-7,-7) $
- $ \triangle P'''Q'''R''' $ coordinates: $ P'''(5,-3) $, $ Q'''(5,-9) $, $ R'''(7,-7) $
- The composition $ r_{y-\text{axis}} \circ r_{x-\text{axis}} \circ R_{90^\circ} $ is a $ 90^\circ $ clockwise rotation about the origin (or $ R_{- 90^\circ} $ or $ R_{270^\circ} $ counterclockwise) with the rule $ (x,y)\to(y,-x) $

## Problem 9:
- $ \triangle A'B'C' $ coordinates: $ A'(3,-5) $, $ B'(4,-7) $, $ C'(7,-4) $
- $ \triangle A''B''C'' $ coordinates: $ A''(-3,5) $, $ B''(-4,7) $, $ C''(-7,4) $
- $ \triangle A'''B'''C''' $ coordinates: $ A'''(-5,3) $, $ B'''(-7,4) $, $ C'''(-4,7) $

(Note: For the graphing part, you would plot the points on the given coordinate grids. For example, for $ \triangle PQR $ with $ P(3,5) $, $ Q(9,5) $, $ R(7,7) $, you would find the points on the grid and connect them. Then follow the same process for the other triangles.)

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QUESTION IMAGE

Question

Explanation:

Problem 8 Solution:

Step 1: Find \( \triangle P'Q'R' \) ( \( R_{90^\circ} \) rotation of \( \triangle PQR \))

Step 2: Find \( \triangle P''Q''R'' \) ( \( r_{x-\text{axis}} \) reflection of \( \triangle P'Q'R' \))

Step 3: Find \( \triangle P'''Q'''R''' \) ( \( r_{y-\text{axis}} \) reflection of \( \triangle P''Q''R'' \))

Step 4: Determine the single transformation

Problem 9 Solution:

Step 1: Find \( \triangle A'B'C' \) (Translation \( T_{4,-3} \) of \( \triangle ABC \))

Step 2: Find \( \triangle A''B''C'' \) (Reflection in the origin of \( \triangle A'B'C' \))

Step 3: Find \( \triangle A'''B'''C''' \) (Reflection in the line \( y=-x \) of \( \triangle A''B''C'' \))

Answer:

Problem 8:

Problem 9: