lesson 2 | session 2
3 figure jklmn is reflected across the x-axis to form figure ( jklmn ). tell whether each statement is true or false.
image of coordinate grid with figure ( jklmn )
a. ( l ) is located at ( (-3, 4) ).
b. ( m ) is located 1 unit below the ( x )-axis.
c. the ( y )-coordinate of ( k ) is the opposite of the ( y )-coordinate of ( k ).
d. the line of reflection is the ( y )-axis.
e. ( nm = nm )
f. the ( x )-coordinate of ( n ) is the opposite of the ( x )-coordinate of ( n ).
g. ( mangle j = mangle j )
4 noor draws ( riangle pqr ) in the coordinate plane as shown. then she reflects ( riangle pqr ) across the ( y )-axis to form ( riangle pqr ). doug incorrectly says the coordinates of ( q ) are ( (-3, 2) ). what are the correct coordinates? how might doug have gotten the coordinates ( (-3, 2) )?
image of coordinate grid with ( riangle pqr )
5 the points given below are the vertices of ( riangle vwx ) and its image after a reflection.
( v(2, 1) ), ( w(3, 5) ), ( x(4, 2) )
( v(-2, 1) ), ( w(-3, 5) ), ( x(-4, 2) )
a. what do you notice when you compare corresponding ( x )-coordinates and corresponding ( y )-coordinates of the vertices of ( riangle vwx ) and ( riangle vwx )?

Question

lesson 2 | session 2
3 figure jklmn is reflected across the x-axis to form figure ( jklmn ). tell whether each statement is true or false.
image of coordinate grid with figure ( jklmn )
a. ( l ) is located at ( (-3, 4) ).
b. ( m ) is located 1 unit below the ( x )-axis.
c. the ( y )-coordinate of ( k ) is the opposite of the ( y )-coordinate of ( k ).
d. the line of reflection is the ( y )-axis.
e. ( nm = nm )
f. the ( x )-coordinate of ( n ) is the opposite of the ( x )-coordinate of ( n ).
g. ( mangle j = mangle j )
4 noor draws ( 	riangle pqr ) in the coordinate plane as shown. then she reflects ( 	riangle pqr ) across the ( y )-axis to form ( 	riangle pqr ). doug incorrectly says the coordinates of ( q ) are ( (-3, 2) ). what are the correct coordinates? how might doug have gotten the coordinates ( (-3, 2) )?
image of coordinate grid with ( 	riangle pqr )
5 the points given below are the vertices of ( 	riangle vwx ) and its image after a reflection.
( v(2, 1) ), ( w(3, 5) ), ( x(4, 2) )
( v(-2, 1) ), ( w(-3, 5) ), ( x(-4, 2) )
a. what do you notice when you compare corresponding ( x )-coordinates and corresponding ( y )-coordinates of the vertices of ( 	riangle vwx ) and ( 	riangle vwx )?

Accepted Answer

### Problem 3: Analyzing Reflections of Figure JKLMN

#### Part a: $ L' $ is located at $ (-3, 4) $
- **Step 1: Find original coordinates of $ L $**
  From the graph, $ L $ is at $ (3, 4) $.
- **Step 2: Reflect over $ x $-axis**
  Reflection over $ x $-axis: $ (x, y) \to (x, -y) $.
  So $ L' = (3, -4) $, not $ (-3, 4) $.
  **Answer: False**

#### Part b: $ M' $ is located 1 unit below the $ x $-axis
- **Step 1: Find original coordinates of $ M $**
  From the graph, $ M $ is at $ (4, 1) $.
- **Step 2: Reflect over $ x $-axis**
  $ M' = (4, -1) $, which is 1 unit below $ x $-axis (since $ y = -1 $).
  **Answer: True**

#### Part c: $ y $-coordinate of $ K' $ is opposite of $ K $’s $ y $-coordinate
- **Step 1: Recall reflection over $ x $-axis**
  Rule: $ (x, y) \to (x, -y) $. Thus, $ y $-coordinate of image is $ -y $ (opposite of original $ y $).
  **Answer: True**

#### Part d: Line of reflection is $ y $-axis
- **Step 1: Recall reflection axes**
  Reflection over $ x $-axis (given) changes $ y $-coordinate; reflection over $ y $-axis changes $ x $-coordinate.
  The problem states reflection over $ x $-axis, so line of reflection is $ x $-axis, not $ y $-axis.
  **Answer: False**

#### Part e: $ N'M' = NM $
- **Step 1: Property of reflections**
  Reflections preserve distance (isometric transformation). Thus, $ N'M' = NM $.
  **Answer: True**

#### Part f: $ x $-coordinate of $ N' $ is opposite of $ N $’s $ x $-coordinate
- **Step 1: Find original coordinates of $ N $**
  From the graph, $ N $ is at $ (1, 2) $.
- **Step 2: Reflect over $ x $-axis**
  $ N' = (1, -2) $. $ x $-coordinate remains $ 1 $, not opposite.
  **Answer: False**

#### Part g: $ m\angle J' = m\angle J $
- **Step 1: Property of reflections**
  Reflections preserve angle measures (isometric transformation). Thus, $ m\angle J' = m\angle J $.
  **Answer: True**

### Problem 4: Reflecting $ \triangle PQR $ over $ y $-axis

#### Step 1: Find original coordinates of $ Q $
From the graph, $ Q $ is at $ (-3, -2) $ (wait, no—wait, the graph shows $ Q $ in the third quadrant? Wait, no, the grid: $ Q $ is at $ (-3, -2) $? Wait, no, the user’s graph: $ Q $ is at $ (-3, -2) $? Wait, no, let’s check the grid. The $ x $-axis: left is negative, right positive. $ y $-axis: up positive, down negative. So $ Q $ is at $ (-3, -2) $? Wait, no, the problem says Doug says $ Q' = (-3, 2) $. Wait, reflection over $ y $-axis: $ (x, y) \to (-x, y) $.

Wait, original $ Q $: let’s look at the graph. The triangle $ \triangle PQR $: $ Q $ is at $ (3, -2) $? Wait, no, the grid: $ x $-axis from -6 to 6, $ y $-axis from -6 to 4. $ Q $ is at $ (3, -2) $? Wait, no, the user’s graph: $ Q $ is at $ (-3, -2) $? Wait, the problem says “reflects $ \triangle PQR $ across the $ y $-axis”. So reflection over $ y $-axis: $ (x, y) \to (-x, y) $.

Wait, let’s re-express: If original $ Q $ is at $ (3, -2) $, then $ Q' = (-3, -2) $. But Doug says $ (-3, 2) $. Wait, maybe original $ Q $ is at $ (3, -2) $, and Doug reflected over $ x $-axis instead of $ y $-axis: reflection over $ x $-axis is $ (x, -y) $, so $ (3, -2) \to (3, 2) $? No, Doug’s answer is $ (-3, 2) $. Wait, maybe original $ Q $ is at $ (3, -2) $, and Doug did both: reflected over $ y $-axis and $ x $-axis? No. Wait, let’s check the graph again. The user’s graph: $ Q $ is at $ (-3, -2) $? Wait, the grid: $ x = -3 $, $ y = -2 $. So original $ Q $ is $ (-3, -2) $. Reflection over $ y $-axis: $ (x, y) \to (-x, y) $, so $ Q' = (3, -2) $. But Doug says $ (-3, 2) $. Wait, maybe Doug reflected over $ x $-axis: $ (x, y) \to (x, -y) $, so $ (-3, -2) \to (-3, 2) $. Ah! So:

- **Correct coordinates of $ Q' $:**
  Original $ Q $ is at $ (-3, -2) $. Reflection over $ y $-axis: $ (x, y) \to (-x, y) $, so $ Q' = (3, -2) $.

- **How Doug got $ (-3, 2) $:**
  He likely reflected over the $ x $-axis (which changes $ y $-coordinate sign) instead of the $ y $-axis (which changes $ x $-coordinate sign). So $ (-3, -2) \to (-3, 2) $ (reflection over $ x $-axis).

### Problem 5a: Analyzing Reflections of $ \triangle VWX $

#### Step 1: List coordinates
- $ V(2, 1) \to V'(-2, 1) $
- $ W(3, 5) \to W'(-3, 5) $
- $ X(4, 2) \to X'(-4, 2) $

#### Step 2: Compare coordinates
- $ x $-coordinates: $ 2 \to -2 $, $ 3 \to -3 $, $ 4 \to -4 $ (opposite of original $ x $-coordinates).
- $ y $-coordinates: $ 1 \to 1 $, $ 5 \to 5 $, $ 2 \to 2 $ (same as original $ y $-coordinates).

#### Conclusion
The $ y $-coordinates remain unchanged, and the $ x $-coordinates are the opposites of the original $ x $-coordinates. This indicates a reflection over the $ y $-axis.

### Final Answers

#### Problem 3
a. False
b. True
c. True
d. False
e. True
f. False
g. True

#### Problem 4
- Correct coordinates of $ Q' $: $ (3, -2) $
- Doug’s mistake: He reflected over the $ x $-axis (changing $ y $-coordinate sign) instead of the $ y $-axis (changing $ x $-coordinate sign).

#### Problem 5a
When comparing coordinates, the $ y $-coordinates of the vertices of $ \triangle VWX $ and $ \triangle V'W'X' $ are the same, while the $ x $-coordinates of the image are the opposites of the original $ x $-coordinates. This means the triangle was reflected over the $ y $-axis.

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QUESTION IMAGE

Question

Explanation:

Problem 3: Analyzing Reflections of Figure JKLMN

Part a: \( L' \) is located at \( (-3, 4) \)

Answer:

Part b: \( M' \) is located 1 unit below the \( x \)-axis