lesson 2 | session 2
3 figure jkl mn is reflected across the x-axis to form figure jklmn. tell whether each statement is true or false.
image of coordinate grid with figure jkl mn
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. m∠j = m∠j
table with true/false columns
4 noor draws △pqr in the coordinate plane as shown. then she reflects △pqr across the y-axis to form △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 △pqr
5 the points given below are the vertices of △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 △vwx and △vwx?
b. what is the line of reflection?

Question

Accepted Answer

### Problem 3 (a - g)
#### a.
# Explanation:
## Step1: Recall reflection over x - axis rule
The rule for reflecting a point $(x,y)$ over the $x$ - axis is $(x,-y)$. First, we need to find the coordinates of $L$ from the graph. From the graph, $L$ is at $(-3, - 4)$ (assuming the original coordinates, since reflection over $x$ - axis changes $y$ - coordinate sign). After reflection over $x$ - axis, $L'$ should be $(-3,4)$.
## Step2: Compare with given
The given coordinate of $L'$ is $(-3,4)$, so the statement is True.
# Answer: True

#### b.
# Explanation:
## Step1: Find coordinates of $M$
From the graph, $M$ is at $(4,1)$.
## Step2: Apply reflection over $x$ - axis rule
Using the rule $(x,y)\to(x, - y)$, $M'$ will be $(4,-1)$, which is 1 unit below the $x$ - axis (since $y=-1$). So the statement is True.
# Answer: True

#### c.
# Explanation:
## Step1: Recall reflection over $x$ - axis rule
The rule for reflection over $x$ - axis is $(x,y)\to(x, - y)$. So if $K=(x,y)$, then $K'=(x, - y)$.
## Step2: Analyze $y$ - coordinates
The $y$ - coordinate of $K'$ is $-y$ and the $y$ - coordinate of $K$ is $y$, so the $y$ - coordinate of $K'$ is the opposite of the $y$ - coordinate of $K$. The statement is True.
# Answer: True

#### d.
# Explanation:
## Step1: Recall reflection lines
The figure is reflected over the $x$ - axis (as per the problem statement: "reflected across the $x$ - axis"). The $y$ - axis is a different line of reflection.
## Step2: Conclude
So the line of reflection is not the $y$ - axis. The statement is False.
# Answer: False

#### e.
# Explanation:
## Step1: Recall reflection property
Reflection is a rigid transformation, which means that the length of a segment and the length of its image after reflection are equal. So $N'M'=NM$ because reflection preserves length.
## Step2: Conclude
The statement is True.
# Answer: True

#### f.
# Explanation:
## Step1: Recall reflection over $x$ - axis rule
Reflection over $x$ - axis changes the $y$ - coordinate, not the $x$ - coordinate. So the $x$ - coordinate of $N'$ is the same as the $x$ - coordinate of $N$, not the opposite.
## Step2: Conclude
The statement is False.
# Answer: False

#### g.
# Explanation:
## Step1: Recall reflection property
Reflection is a rigid transformation, which preserves angle measures. So $m\angle J' = m\angle J$.
## Step2: Conclude
The statement is True.
# Answer: True

### Problem 4
# Explanation:
## Step1: Recall reflection over $y$ - axis rule
The rule for reflecting a point $(x,y)$ over the $y$ - axis is $(x,y)\to(-x,y)$. First, we need to find the coordinates of $Q$ from the graph. From the graph, $Q$ is at $(3, - 2)$ (assuming the position of $Q$ in the coordinate plane).
## Step2: Apply the reflection rule
Using the rule $(x,y)\to(-x,y)$, for $Q=(3,-2)$, $Q'$ should be $(-3,-2)$.
## Step3: Analyze Doug's mistake
Doug got $(-3,2)$. He might have reflected over the $x$ - axis instead of the $y$ - axis. The rule for reflection over $x$ - axis is $(x,y)\to(x, - y)$. If he applied the $x$ - axis reflection rule to $Q=(3,-2)$, he would get $(3,2)$ (which is wrong), but maybe he misread the coordinates of $Q$ or misapplied the reflection axis. Another possibility is that he thought the reflection was over the $x$ - axis and also changed the $x$ - coordinate sign, but the correct reflection over $y$ - axis only changes the $x$ - coordinate sign and keeps the $y$ - coordinate the same.
# Answer: The correct coordinates of $Q'$ are $(-3,-2)$. Doug might have reflected the point over the $x$ - axis (used the rule $(x,y)\to(x, - y)$) instead of the $y$ - axis (rule $(x,y)\to(-x,y)$) or misread the original $y$ - coordinate of $Q$.

### Problem 5
#### a.
# Explanation:
## Step1: List coordinates
For $\triangle VWX$: $V(2,1)$, $W(3,5)$, $X(4,2)$
For $\triangle V'W'X'$: $V'(-2,1)$, $W'(-3,5)$, $X'(-4,2)$
## Step2: Compare $x$ and $y$ - coordinates
Compare the $x$ - coordinates: $2$ and $-2$ (opposites), $3$ and $-3$ (opposites), $4$ and $-4$ (opposites). Compare the $y$ - coordinates: $1 = 1$, $5 = 5$, $2 = 2$. So the $x$ - coordinates of the image are the opposites of the $x$ - coordinates of the original triangle, and the $y$ - coordinates are the same.
# Answer: The $y$ - coordinates of the corresponding vertices are equal, and the $x$ - coordinates of the image vertices are the opposites of the $x$ - coordinates of the original vertices.

#### b.
# Explanation:
## Step1: Recall reflection rules
When a point $(x,y)$ is reflected over the $y$ - axis, the rule is $(x,y)\to(-x,y)$, which matches the pattern we saw in part (a) ( $x$ - coordinates are opposites, $y$ - coordinates are same).
## Step2: Conclude the line of reflection
So the line of reflection is the $y$ - axis ( $x = 0$).
# Answer: The line of reflection is the $y$ - axis ( $x=0$).

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QUESTION IMAGE

Question

Explanation:

Problem 3 (a - g)

a.

Step1: Recall reflection over x - axis rule

Step2: Compare with given

Step1: Find coordinates of \(M\)

Step2: Apply reflection over \(x\) - axis rule

Step1: Recall reflection over \(x\) - axis rule

Step2: Analyze \(y\) - coordinates

Answer:

b.