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.

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.
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.

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:

\[
\boxed{(7, 3)}
\]
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'' = \( \boxed{(0, 1)} \)
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 \( A'B'C' \).

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