Question
Triangle QRS has coordinates Q(-8,4), R(-5,8) and S(-3,3). Triangle QRS is transformed by (x,y)→(x+4,y−6)
(
𝑥
,
𝑦
)
→
(
𝑥
+
4
,
𝑦
−
6
)
to produce △Q′R′S′
△
𝑄
′
𝑅
′
𝑆
′
. Then △Q′R′S′
△
𝑄
′
𝑅
′
𝑆
′
is transformed by the rule (x,y)→(x+5,y−3)
(
𝑥
,
𝑦
)
→
(
𝑥
+
5
,
𝑦
−
3
)
to produce △Q"R"S"
△
𝑄
"
𝑅
"
𝑆
"
.
a. The final image, △Q"R"S"
△
𝑄
"
𝑅
"
𝑆
"
, will be in which quadrant?
b. Create the rule for the composition transformation. example (x+2,y-4)
(
𝑥
,
𝑦
)
→
(
𝑥
+
4
,
𝑦
−
6
)
to produce △Q′R′S′
△
𝑄
′
𝑅
′
𝑆
′
. Then △Q′R′S′
△
𝑄
′
𝑅
′
𝑆
′
is transformed by the rule (x,y)→(x+5,y−3)
(
𝑥
,
𝑦
)
→
(
𝑥
+
5
,
𝑦
−
3
)
to produce △Q"R"S"
△
𝑄
"
𝑅
"
𝑆
"
.
a. The final image, △Q"R"S"
△
𝑄
"
𝑅
"
𝑆
"
, will be in which quadrant?
b. Create the rule for the composition transformation. example (x+2,y-4)
Answers
GPT-4o mini
To resolve the transformations and the quadrants of the final image of triangle \( QRS \), we need to carry out each transformation step by step.
### Step 1: First Transformation
Triangle \( QRS \) has the following vertices:
- \( Q(-8, 4) \)
- \( R(-5, 8) \)
- \( S(-3, 3) \)
The first transformation is given by the rule \( (x, y) \to (x + 4, y - 6) \).
Applying this transformation to each vertex:
- For \( Q(-8, 4) \):
\[
Q' = (-8 + 4, 4 - 6) = (-4, -2)
\]
- For \( R(-5, 8) \):
\[
R' = (-5 + 4, 8 - 6) = (-1, 2)
\]
- For \( S(-3, 3) \):
\[
S' = (-3 + 4, 3 - 6) = (1, -3)
\]
So, after the first transformation, the coordinates of triangle \( Q'R'S' \) are:
- \( Q'(-4, -2) \)
- \( R'(-1, 2) \)
- \( S'(1, -3) \)
### Step 2: Second Transformation
Now applying the second transformation, \( (x, y) \to (x + 5, y - 3) \), to each vertex \( Q'R'S' \):
- For \( Q'(-4, -2) \):
\[
Q'' = (-4 + 5, -2 - 3) = (1, -5)
\]
- For \( R'(-1, 2) \):
\[
R'' = (-1 + 5, 2 - 3) = (4, -1)
\]
- For \( S'(1, -3) \):
\[
S'' = (1 + 5, -3 - 3) = (6, -6)
\]
Thus, the coordinates of triangle \( Q''R''S'' \) are:
- \( Q''(1, -5) \)
- \( R''(4, -1) \)
- \( S''(6, -6) \)
### a. Determining the Quadrant
The points \( Q''(1, -5) \), \( R''(4, -1) \), and \( S''(6, -6) \) all have a positive x-coordinate and a negative y-coordinate. Therefore, they are located in the **fourth quadrant**.
### b. Creating the Rule for the Composition Transformation
The first transformation is \( (x, y) \to (x + 4, y - 6) \) and the second transformation is \( (x, y) \to (x + 5, y - 3) \).
To create the composition, we add the changes:
1. From the first transformation, we add 4 to the x-coordinate and subtract 6 from the y-coordinate.
2. From the second transformation, we add 5 to the x-coordinate and subtract 3 from the y-coordinate.
Combining these gives:
\[
x \to x + 4 + 5 = x + 9
\]
\[
y \to y - 6 - 3 = y - 9
\]
Thus, the rule for the composition transformation is:
\[
(x, y) \to (x + 9, y - 9)
\]
### Step 1: First Transformation
Triangle \( QRS \) has the following vertices:
- \( Q(-8, 4) \)
- \( R(-5, 8) \)
- \( S(-3, 3) \)
The first transformation is given by the rule \( (x, y) \to (x + 4, y - 6) \).
Applying this transformation to each vertex:
- For \( Q(-8, 4) \):
\[
Q' = (-8 + 4, 4 - 6) = (-4, -2)
\]
- For \( R(-5, 8) \):
\[
R' = (-5 + 4, 8 - 6) = (-1, 2)
\]
- For \( S(-3, 3) \):
\[
S' = (-3 + 4, 3 - 6) = (1, -3)
\]
So, after the first transformation, the coordinates of triangle \( Q'R'S' \) are:
- \( Q'(-4, -2) \)
- \( R'(-1, 2) \)
- \( S'(1, -3) \)
### Step 2: Second Transformation
Now applying the second transformation, \( (x, y) \to (x + 5, y - 3) \), to each vertex \( Q'R'S' \):
- For \( Q'(-4, -2) \):
\[
Q'' = (-4 + 5, -2 - 3) = (1, -5)
\]
- For \( R'(-1, 2) \):
\[
R'' = (-1 + 5, 2 - 3) = (4, -1)
\]
- For \( S'(1, -3) \):
\[
S'' = (1 + 5, -3 - 3) = (6, -6)
\]
Thus, the coordinates of triangle \( Q''R''S'' \) are:
- \( Q''(1, -5) \)
- \( R''(4, -1) \)
- \( S''(6, -6) \)
### a. Determining the Quadrant
The points \( Q''(1, -5) \), \( R''(4, -1) \), and \( S''(6, -6) \) all have a positive x-coordinate and a negative y-coordinate. Therefore, they are located in the **fourth quadrant**.
### b. Creating the Rule for the Composition Transformation
The first transformation is \( (x, y) \to (x + 4, y - 6) \) and the second transformation is \( (x, y) \to (x + 5, y - 3) \).
To create the composition, we add the changes:
1. From the first transformation, we add 4 to the x-coordinate and subtract 6 from the y-coordinate.
2. From the second transformation, we add 5 to the x-coordinate and subtract 3 from the y-coordinate.
Combining these gives:
\[
x \to x + 4 + 5 = x + 9
\]
\[
y \to y - 6 - 3 = y - 9
\]
Thus, the rule for the composition transformation is:
\[
(x, y) \to (x + 9, y - 9)
\]