Question
Rectangle PQRS
has vertices P(−3,3)
, Q(6,3)
, R(6,−3)
, and S(−3,−3)
. Graph the rectangle and its image after a dilation with a scale factor of 13
.
has vertices P(−3,3)
, Q(6,3)
, R(6,−3)
, and S(−3,−3)
. Graph the rectangle and its image after a dilation with a scale factor of 13
.
Answers
GPT-4o mini
To graph the rectangle PQRS and its image after a dilation, we will first identify the original vertices of the rectangle and then apply the dilation transformation.
**Original Rectangle PQRS:**
- P(-3, 3)
- Q(6, 3)
- R(6, -3)
- S(-3, -3)
**Graphing PQRS:**
1. Plot the points P, Q, R, and S on a coordinate system.
2. Connect the points in the order: P to Q, Q to R, R to S, and S back to P to complete the rectangle.
**Dilation with a Scale Factor of 13:**
To find the image after dilation, we first need to determine a center of dilation. Typically, for this kind of problem, we assume the center of dilation is at the origin (0, 0).
To dilate each point (x, y) using a scale factor \( k \), we use the formula:
\[
(x', y') = (k \cdot x, k \cdot y)
\]
In this case, \( k = 13 \).
Now let's calculate the new coordinates for each vertex after dilation.
1. **For P(-3, 3)**:
\[
P' = (13 \cdot -3, 13 \cdot 3) = (-39, 39)
\]
2. **For Q(6, 3)**:
\[
Q' = (13 \cdot 6, 13 \cdot 3) = (78, 39)
\]
3. **For R(6, -3)**:
\[
R' = (13 \cdot 6, 13 \cdot -3) = (78, -39)
\]
4. **For S(-3, -3)**:
\[
S' = (13 \cdot -3, 13 \cdot -3) = (-39, -39)
\]
**Dilated Rectangle P'Q'R'S':**
- P'(-39, 39)
- Q'(78, 39)
- R'(78, -39)
- S'(-39, -39)
**Graphing P'Q'R'S':**
1. Plot the points P', Q', R', and S' on the same coordinate system.
2. Connect the points in the order: P' to Q', Q' to R', R' to S', and S' back to P' to complete the dilation of the rectangle.
### Summary
- The original rectangle PQRS has vertices at P(-3, 3), Q(6, 3), R(6, -3), S(-3, -3).
- After dilation by a scale factor of 13, the vertices of the rectangle become P'(-39, 39), Q'(78, 39), R'(78, -39), S'(-39, -39).
You can visualize the original rectangle being significantly expanded, so the new rectangle will lie far away from the original rectangle in the coordinate plane.
**Original Rectangle PQRS:**
- P(-3, 3)
- Q(6, 3)
- R(6, -3)
- S(-3, -3)
**Graphing PQRS:**
1. Plot the points P, Q, R, and S on a coordinate system.
2. Connect the points in the order: P to Q, Q to R, R to S, and S back to P to complete the rectangle.
**Dilation with a Scale Factor of 13:**
To find the image after dilation, we first need to determine a center of dilation. Typically, for this kind of problem, we assume the center of dilation is at the origin (0, 0).
To dilate each point (x, y) using a scale factor \( k \), we use the formula:
\[
(x', y') = (k \cdot x, k \cdot y)
\]
In this case, \( k = 13 \).
Now let's calculate the new coordinates for each vertex after dilation.
1. **For P(-3, 3)**:
\[
P' = (13 \cdot -3, 13 \cdot 3) = (-39, 39)
\]
2. **For Q(6, 3)**:
\[
Q' = (13 \cdot 6, 13 \cdot 3) = (78, 39)
\]
3. **For R(6, -3)**:
\[
R' = (13 \cdot 6, 13 \cdot -3) = (78, -39)
\]
4. **For S(-3, -3)**:
\[
S' = (13 \cdot -3, 13 \cdot -3) = (-39, -39)
\]
**Dilated Rectangle P'Q'R'S':**
- P'(-39, 39)
- Q'(78, 39)
- R'(78, -39)
- S'(-39, -39)
**Graphing P'Q'R'S':**
1. Plot the points P', Q', R', and S' on the same coordinate system.
2. Connect the points in the order: P' to Q', Q' to R', R' to S', and S' back to P' to complete the dilation of the rectangle.
### Summary
- The original rectangle PQRS has vertices at P(-3, 3), Q(6, 3), R(6, -3), S(-3, -3).
- After dilation by a scale factor of 13, the vertices of the rectangle become P'(-39, 39), Q'(78, 39), R'(78, -39), S'(-39, -39).
You can visualize the original rectangle being significantly expanded, so the new rectangle will lie far away from the original rectangle in the coordinate plane.