Asked by NICKOLAS
Triangle JKL
has vertices J(3, –7)
, K(7, –3)
and L(3, 7)
. What are the coordinates of J
after a 90°
rotation counterclockwise about the origin? Enter the answer.
has vertices J(3, –7)
, K(7, –3)
and L(3, 7)
. What are the coordinates of J
after a 90°
rotation counterclockwise about the origin? Enter the answer.
Answers
Answered by
NICKOLAS
Figure ABCD
has vertices A(–2, –2)
, B(1, 1)
, C(0, 4)
, and D(–3, 5)
. The figure is translated 5
units to the right and 6
units down. Then it is translated 2
units to the left and 2
units up. What are the coordinates of image A″B″C″D″
after both translations? Enter the answers.
has vertices A(–2, –2)
, B(1, 1)
, C(0, 4)
, and D(–3, 5)
. The figure is translated 5
units to the right and 6
units down. Then it is translated 2
units to the left and 2
units up. What are the coordinates of image A″B″C″D″
after both translations? Enter the answers.
Answered by
NICKOLAS
Triangle ABC has coordinates A(3, 3) , B(5, –2) , and C(4, 4) . Graph the preimage and image of the triangle after it is reflected across the line x=1 .
Explain how to write the coordinates of the vertices of the reflected triangle. Enter the answer.
Explain how to write the coordinates of the vertices of the reflected triangle. Enter the answer.
Answered by
GPT-4o mini
To perform a 90° counterclockwise rotation about the origin, you can use the transformation:
\[
(x, y) \rightarrow (-y, x)
\]
Starting with the coordinates of point J, which are \( J(3, -7) \):
1. Take the \( y \)-coordinate: \(-7\)
2. Take the \( x \)-coordinate: \(3\)
Now apply the transformation:
\[
J'(x', y') = (-(-7), 3) = (7, 3)
\]
So the coordinates of point J after a 90° rotation counterclockwise about the origin are:
\[
\boxed{(7, 3)}
\]
\[
(x, y) \rightarrow (-y, x)
\]
Starting with the coordinates of point J, which are \( J(3, -7) \):
1. Take the \( y \)-coordinate: \(-7\)
2. Take the \( x \)-coordinate: \(3\)
Now apply the transformation:
\[
J'(x', y') = (-(-7), 3) = (7, 3)
\]
So the coordinates of point J after a 90° rotation counterclockwise about the origin are:
\[
\boxed{(7, 3)}
\]
Answered by
GPT-4o mini
Let's perform the translations step-by-step for each vertex of the figure ABCD.
### Step 1: Initial Coordinates
- A: (−2, −2)
- B: (1, 1)
- C: (0, 4)
- D: (−3, 5)
### Step 2: First Translation
The first translation is 5 units to the right and 6 units down. This means we add 5 to the x-coordinate and subtract 6 from the y-coordinate.
#### Translations:
1. A(−2, −2):
\[
A' = (−2 + 5, −2 - 6) = (3, -8)
\]
2. B(1, 1):
\[
B' = (1 + 5, 1 - 6) = (6, -5)
\]
3. C(0, 4):
\[
C' = (0 + 5, 4 - 6) = (5, -2)
\]
4. D(−3, 5):
\[
D' = (−3 + 5, 5 - 6) = (2, -1)
\]
### Step 3: Second Translation
The second translation is 2 units to the left and 2 units up. This means we subtract 2 from the x-coordinate and add 2 to the y-coordinate.
#### Translations:
1. A'(3, −8):
\[
A'' = (3 - 2, -8 + 2) = (1, -6)
\]
2. B'(6, −5):
\[
B'' = (6 - 2, -5 + 2) = (4, -3)
\]
3. C'(5, −2):
\[
C'' = (5 - 2, -2 + 2) = (3, 0)
\]
4. D'(2, −1):
\[
D'' = (2 - 2, -1 + 2) = (0, 1)
\]
### Final Coordinates of A'', B'', C'', and D''
The coordinates of the image \( A''B''C''D'' \) after the two translations are:
- A'': \( (1, -6) \)
- B'': \( (4, -3) \)
- C'': \( (3, 0) \)
- D'': \( (0, 1) \)
So, the answers are:
- A'' = \( (1, -6) \)
- B'' = \( (4, -3) \)
- C'' = \( (3, 0) \)
- D'' = \( (0, 1) \)
You can represent this as:
- A'' = \( \boxed{(1, -6)} \)
- B'' = \( \boxed{(4, -3)} \)
- C'' = \( \boxed{(3, 0)} \)
- D'' = \( \boxed{(0, 1)} \)
### Step 1: Initial Coordinates
- A: (−2, −2)
- B: (1, 1)
- C: (0, 4)
- D: (−3, 5)
### Step 2: First Translation
The first translation is 5 units to the right and 6 units down. This means we add 5 to the x-coordinate and subtract 6 from the y-coordinate.
#### Translations:
1. A(−2, −2):
\[
A' = (−2 + 5, −2 - 6) = (3, -8)
\]
2. B(1, 1):
\[
B' = (1 + 5, 1 - 6) = (6, -5)
\]
3. C(0, 4):
\[
C' = (0 + 5, 4 - 6) = (5, -2)
\]
4. D(−3, 5):
\[
D' = (−3 + 5, 5 - 6) = (2, -1)
\]
### Step 3: Second Translation
The second translation is 2 units to the left and 2 units up. This means we subtract 2 from the x-coordinate and add 2 to the y-coordinate.
#### Translations:
1. A'(3, −8):
\[
A'' = (3 - 2, -8 + 2) = (1, -6)
\]
2. B'(6, −5):
\[
B'' = (6 - 2, -5 + 2) = (4, -3)
\]
3. C'(5, −2):
\[
C'' = (5 - 2, -2 + 2) = (3, 0)
\]
4. D'(2, −1):
\[
D'' = (2 - 2, -1 + 2) = (0, 1)
\]
### Final Coordinates of A'', B'', C'', and D''
The coordinates of the image \( A''B''C''D'' \) after the two translations are:
- A'': \( (1, -6) \)
- B'': \( (4, -3) \)
- C'': \( (3, 0) \)
- D'': \( (0, 1) \)
So, the answers are:
- A'' = \( (1, -6) \)
- B'' = \( (4, -3) \)
- C'' = \( (3, 0) \)
- D'' = \( (0, 1) \)
You can represent this as:
- A'' = \( \boxed{(1, -6)} \)
- B'' = \( \boxed{(4, -3)} \)
- C'' = \( \boxed{(3, 0)} \)
- D'' = \( \boxed{(0, 1)} \)
Answered by
GPT-4o mini
To reflect the triangle ABC across the line \( x = 1 \), we need to determine how the coordinates of each vertex transform under this reflection.
### Reflection Rule:
When reflecting a point \((x, y)\) across the vertical line \(x = k\), the new x-coordinate can be calculated using the formula:
\[
x' = 2k - x
\]
The y-coordinate remains unchanged:
\[
y' = y
\]
In this case, the line is \(x = 1\). Therefore, for each vertex \((x, y)\), the reflected coordinates will be:
\[
x' = 2(1) - x = 2 - x
\]
\[
y' = y
\]
### Step 1: Reflect each vertex of the triangle ABC
1. **Vertex A(3, 3)**:
- \( x' = 2 - 3 = -1 \)
- \( y' = 3 \)
- Thus, \( A' = (-1, 3) \)
2. **Vertex B(5, -2)**:
- \( x' = 2 - 5 = -3 \)
- \( y' = -2 \)
- Thus, \( B' = (-3, -2) \)
3. **Vertex C(4, 4)**:
- \( x' = 2 - 4 = -2 \)
- \( y' = 4 \)
- Thus, \( C' = (-2, 4) \)
### Step 2: Summarize the coordinates of the reflected triangle
The coordinates of the vertices of the reflected triangle \( A'B'C' \) are:
- \( A' = (-1, 3) \)
- \( B' = (-3, -2) \)
- \( C' = (-2, 4) \)
### Result:
So, the coordinates of the vertices of the reflected triangle are:
- \( A' = \boxed{(-1, 3)} \)
- \( B' = \boxed{(-3, -2)} \)
- \( C' = \boxed{(-2, 4)} \)
### Explanation:
To reflect the vertices of triangle ABC across the line \( x = 1 \), we determined the new x-coordinates by applying the reflection formula while keeping the y-coordinates unchanged. The resulting coordinates represent the vertices of the reflected triangle \( A'B'C' \).
### Reflection Rule:
When reflecting a point \((x, y)\) across the vertical line \(x = k\), the new x-coordinate can be calculated using the formula:
\[
x' = 2k - x
\]
The y-coordinate remains unchanged:
\[
y' = y
\]
In this case, the line is \(x = 1\). Therefore, for each vertex \((x, y)\), the reflected coordinates will be:
\[
x' = 2(1) - x = 2 - x
\]
\[
y' = y
\]
### Step 1: Reflect each vertex of the triangle ABC
1. **Vertex A(3, 3)**:
- \( x' = 2 - 3 = -1 \)
- \( y' = 3 \)
- Thus, \( A' = (-1, 3) \)
2. **Vertex B(5, -2)**:
- \( x' = 2 - 5 = -3 \)
- \( y' = -2 \)
- Thus, \( B' = (-3, -2) \)
3. **Vertex C(4, 4)**:
- \( x' = 2 - 4 = -2 \)
- \( y' = 4 \)
- Thus, \( C' = (-2, 4) \)
### Step 2: Summarize the coordinates of the reflected triangle
The coordinates of the vertices of the reflected triangle \( A'B'C' \) are:
- \( A' = (-1, 3) \)
- \( B' = (-3, -2) \)
- \( C' = (-2, 4) \)
### Result:
So, the coordinates of the vertices of the reflected triangle are:
- \( A' = \boxed{(-1, 3)} \)
- \( B' = \boxed{(-3, -2)} \)
- \( C' = \boxed{(-2, 4)} \)
### Explanation:
To reflect the vertices of triangle ABC across the line \( x = 1 \), we determined the new x-coordinates by applying the reflection formula while keeping the y-coordinates unchanged. The resulting coordinates represent the vertices of the reflected triangle \( A'B'C' \).