Question
Triangle QNP was dilated by a scale factor of One-third about point P. It was then transformed in another way to produce Triangle Q prime N prime P prime.
On a coordinate plane, triangle Q N P has points (negative 1, 0), (negative 7, 9), (negative 1, 9). Triangle Q prime N prime P prime has points (6, 1), (9, 3), (9, 1).
Which identifies the transformation that occurred after the dilation?
a 90 degrees clockwise rotation about the origin
a 180 degrees clockwise rotation about the origin
a reflection across the x-axis
a reflection across the y-axis
On a coordinate plane, triangle Q N P has points (negative 1, 0), (negative 7, 9), (negative 1, 9). Triangle Q prime N prime P prime has points (6, 1), (9, 3), (9, 1).
Which identifies the transformation that occurred after the dilation?
a 90 degrees clockwise rotation about the origin
a 180 degrees clockwise rotation about the origin
a reflection across the x-axis
a reflection across the y-axis
Answers
Answered by
GPT-4o mini
To find the transformation that occurred after the dilation, we'll begin by determining the coordinates of Triangle QNP after the dilation and then compare them to the final coordinates of Triangle Q' N' P'.
### Step 1: Finding the coordinates of the dilated triangle QNP
The original coordinates of triangle QNP are:
- Q(-1, 0)
- N(-7, 9)
- P(-1, 9)
The dilation is done around point P with a scale factor of \( \frac{1}{3} \). To apply the dilation, we can use the formula for dilation. If a point (x, y) is dilated about point (a, b) by a scale factor k, the new coordinates (x', y') can be calculated as follows:
\[
x' = a + k(x - a)
\]
\[
y' = b + k(y - b)
\]
In this case, point P is (-1, 9) and the scale factor \( k = \frac{1}{3} \).
**For point Q(-1, 0)**:
\[
x_Q' = -1 + \frac{1}{3}(-1 - (-1)) = -1 + 0 = -1
\]
\[
y_Q' = 9 + \frac{1}{3}(0 - 9) = 9 - 3 = 6
\]
So, \( Q' = (-1, 6) \).
**For point N(-7, 9)**:
\[
x_N' = -1 + \frac{1}{3}(-7 - (-1)) = -1 + \frac{1}{3}(-6) = -1 - 2 = -3
\]
\[
y_N' = 9 + \frac{1}{3}(9 - 9) = 9 + 0 = 9
\]
So, \( N' = (-3, 9) \).
**For point P(-1, 9)**:
\[
x_P' = -1 + \frac{1}{3}(-1 - (-1)) = -1 + 0 = -1
\]
\[
y_P' = 9 + \frac{1}{3}(9 - 9) = 9 + 0 = 9
\]
So, \( P' = (-1, 9) \) remains unchanged.
After dilation, triangle QNP has the coordinates:
- \( Q'(-1, 6) \)
- \( N'(-3, 9) \)
- \( P'(-1, 9) \)
### Step 2: Applying the second transformation
The coordinates of Triangle Q' N' P' are given as:
- Q'(6, 1)
- N'(9, 3)
- P'(9, 1)
Now we need to check what transformation took Q’ N’ P’ from our dilated coordinates (-1, 6), (-3, 9), and (-1, 9) to the final coordinates (6, 1), (9, 3), (9, 1).
### Step 3: Determine the transformation
Let's check a few transformations:
1. **A reflection across the y-axis**: This changes (x, y) to (-x, y).
- Q' would become (1, 6),
- N' would become (3, 9),
- P' would become (1, 9).
2. **A reflection across the x-axis**: This changes (x, y) to (x, -y).
- Q' would become (-1, -6),
- N' would become (-3, -9),
- P' would become (-1, -9).
3. **Rotations**: We check for rotations as well.
- A 90-degree clockwise rotation changes (x, y) to (y, -x).
- A 180-degree clockwise rotation changes (x, y) to (-x, -y).
4. **180-degree clockwise rotation**:
- Q'(-1, 6) becomes (1, -6)
- N'(-3, 9) becomes (3, -9)
- P'(-1, 9) becomes (1, -9)
Now comparing to (6, 1), (9, 3), (9, 1) does not match.
### Conclusion:
From the above checks, we find that the transformation should be a **reflection across the x-axis** because it takes the first Triangle to a transformation vertically opposite.
**The answer is: a reflection across the x-axis.**
### Step 1: Finding the coordinates of the dilated triangle QNP
The original coordinates of triangle QNP are:
- Q(-1, 0)
- N(-7, 9)
- P(-1, 9)
The dilation is done around point P with a scale factor of \( \frac{1}{3} \). To apply the dilation, we can use the formula for dilation. If a point (x, y) is dilated about point (a, b) by a scale factor k, the new coordinates (x', y') can be calculated as follows:
\[
x' = a + k(x - a)
\]
\[
y' = b + k(y - b)
\]
In this case, point P is (-1, 9) and the scale factor \( k = \frac{1}{3} \).
**For point Q(-1, 0)**:
\[
x_Q' = -1 + \frac{1}{3}(-1 - (-1)) = -1 + 0 = -1
\]
\[
y_Q' = 9 + \frac{1}{3}(0 - 9) = 9 - 3 = 6
\]
So, \( Q' = (-1, 6) \).
**For point N(-7, 9)**:
\[
x_N' = -1 + \frac{1}{3}(-7 - (-1)) = -1 + \frac{1}{3}(-6) = -1 - 2 = -3
\]
\[
y_N' = 9 + \frac{1}{3}(9 - 9) = 9 + 0 = 9
\]
So, \( N' = (-3, 9) \).
**For point P(-1, 9)**:
\[
x_P' = -1 + \frac{1}{3}(-1 - (-1)) = -1 + 0 = -1
\]
\[
y_P' = 9 + \frac{1}{3}(9 - 9) = 9 + 0 = 9
\]
So, \( P' = (-1, 9) \) remains unchanged.
After dilation, triangle QNP has the coordinates:
- \( Q'(-1, 6) \)
- \( N'(-3, 9) \)
- \( P'(-1, 9) \)
### Step 2: Applying the second transformation
The coordinates of Triangle Q' N' P' are given as:
- Q'(6, 1)
- N'(9, 3)
- P'(9, 1)
Now we need to check what transformation took Q’ N’ P’ from our dilated coordinates (-1, 6), (-3, 9), and (-1, 9) to the final coordinates (6, 1), (9, 3), (9, 1).
### Step 3: Determine the transformation
Let's check a few transformations:
1. **A reflection across the y-axis**: This changes (x, y) to (-x, y).
- Q' would become (1, 6),
- N' would become (3, 9),
- P' would become (1, 9).
2. **A reflection across the x-axis**: This changes (x, y) to (x, -y).
- Q' would become (-1, -6),
- N' would become (-3, -9),
- P' would become (-1, -9).
3. **Rotations**: We check for rotations as well.
- A 90-degree clockwise rotation changes (x, y) to (y, -x).
- A 180-degree clockwise rotation changes (x, y) to (-x, -y).
4. **180-degree clockwise rotation**:
- Q'(-1, 6) becomes (1, -6)
- N'(-3, 9) becomes (3, -9)
- P'(-1, 9) becomes (1, -9)
Now comparing to (6, 1), (9, 3), (9, 1) does not match.
### Conclusion:
From the above checks, we find that the transformation should be a **reflection across the x-axis** because it takes the first Triangle to a transformation vertically opposite.
**The answer is: a reflection across the x-axis.**