https://file.aiquickdraw.com/mathbot/file/1726617490841smigupk5.jpg

The problem involves a triangle \( PQR \) with vertices \( P(-4, 9) \), \( Q(-3, 3) \), and \( R(3, 10) \), and asks to apply a dilation with a scale factor of \( k = 2 \) and a center of dilation at \( (-2, 7) \).

To apply the dilation, we use the formula:

\[
P' = C + k \cdot (P - C)
\]

Where:
- \( P \) is the original point (one of the vertices),
- \( C \) is the center of dilation, which is \( (-2, 7) \),
- \( k \) is the scale factor, here \( k = 2 \),
- \( P' \) is the new point after dilation.

Let’s calculate the dilated coordinates for each vertex \( P \), \( Q \), and \( R \).

### For point \( P(-4, 9) \):
1. Subtract \( C(-2, 7) \) from \( P(-4, 9) \):  
   \[ (-4 - (-2), 9 - 7) = (-2, 2) \]
2. Multiply by the scale factor \( k = 2 \):  
   \[ 2 \times (-2, 2) = (-4, 4) \]
3. Add \( C(-2, 7) \) to get the new coordinates:  
   \[ (-4 + (-2), 4 + 7) = (-6, 11) \]
So, \( P' = (-6, 11) \).

### For point \( Q(-3, 3) \):
1. Subtract \( C(-2, 7) \) from \( Q(-3, 3) \):  
   \[ (-3 - (-2), 3 - 7) = (-1, -4) \]
2. Multiply by the scale factor \( k = 2 \):  
   \[ 2 \times (-1, -4) = (-2, -8) \]
3. Add \( C(-2, 7) \) to get the new coordinates:  
   \[ (-2 + (-2), -8 + 7) = (-4, -1) \]
So, \( Q' = (-4, -1) \).

### For point \( R(3, 10) \):
1. Subtract \( C(-2, 7) \) from \( R(3, 10) \):  
   \[ (3 - (-2), 10 - 7) = (5, 3) \]
2. Multiply by the scale factor \( k = 2 \):  
   \[ 2 \times (5, 3) = (10, 6) \]
3. Add \( C(-2, 7) \) to get the new coordinates:  
   \[ (10 + (-2), 6 + 7) = (8, 13) \]
So, \( R' = (8, 13) \).

### Final Dilated Coordinates:
- \( P' = (-6, 11) \)
- \( Q' = (-4, -1) \)
- \( R' = (8, 13) \)

Let me know if you want further clarifications!

Here are five related questions:
1. How would the dilation change if the scale factor were \( \frac{1}{2} \)?
2. What happens if the center of dilation moves to the origin?
3. Can you calculate the area of the original and dilated triangles?
4. How would a reflection across the x-axis affect the dilated triangle?
5. What are the coordinates if we perform a rotation of 90° counterclockwise on the dilated triangle?

**Tip**: When performing dilations, always subtract the center of dilation first before scaling!

Triangle PQR with vertices P(-4, 9), Q(-3, 3), and R(3, 10); scale factor: k = 2, center of dilation: (-2, 7)

This problem involves dilating triangle PQR with vertices P(-4, 9), Q(-3, 3), and R(3, 10) using a scale factor of 2 and a center of dilation at (-2, 7). Learn how to apply the dilation formula to find the new coordinates of the triangle.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation