Question

Use the given vertices to graph ABCD and its image after a dilation centered at the origin with scale factor k = -1/5
B(-5, -10), C(-10, 15), D(0, 5)

Answer 1

To graph the vertices of quadrilateral ABCD and its dilated image centered at the origin with a scale factor of \( k = -\frac{1}{5} \), we first need to identify the coordinates of vertex A. Since the coordinates for points B, C, and D are provided, we will assume a hypothetical coordinate for point A to create a complete quadrilateral.

### Given Vertices:
- B(-5, -10)
- C(-10, 15)
- D(0, 5)

For the sake of simplicity, let's assume point A is at (0, 0). You can use any other values for point A as needed.

### Step 1: Plotting the Original Quadrilateral ABCD
1. **Plot the points on a graph:**
- Point A(0, 0)
- Point B(-5, -10)
- Point C(-10, 15)
- Point D(0, 5)

2. **Connect the points to form quadrilateral ABCD:**
- Connect A to B
- Connect B to C
- Connect C to D
- Connect D to A

### Step 2: Dilating the Points
To find the new coordinates after the dilation with a scale factor of \( k = -\frac{1}{5} \), use the formula:
\[
(x', y') = (k \cdot x, k \cdot y)
\]

#### Calculate the new coordinates for each point:
1. **Point A(0, 0):**
\[
A' = \left(-\frac{1}{5} \cdot 0, -\frac{1}{5} \cdot 0\right) = (0, 0)
\]

2. **Point B(-5, -10):**
\[
B' = \left(-\frac{1}{5} \cdot -5, -\frac{1}{5} \cdot -10\right) = \left(1, 2\right)
\]

3. **Point C(-10, 15):**
\[
C' = \left(-\frac{1}{5} \cdot -10, -\frac{1}{5} \cdot 15\right) = \left(2, -3\right)
\]

4. **Point D(0, 5):**
\[
D' = \left(-\frac{1}{5} \cdot 0, -\frac{1}{5} \cdot 5\right) = \left(0, -1\right)
\]

### Step 3: Plotting the Dilated Quadrilateral A'B'C'D'
1. **Plot the new points:**
- A'(0, 0)
- B'(1, 2)
- C'(2, -3)
- D'(0, -1)

2. **Connect the points to form quadrilateral A'B'C'D':**
- Connect A' to B'
- Connect B' to C'
- Connect C' to D'
- Connect D' to A'

### Final Graph
You should end up with two quadrilaterals:
- The original quadrilateral ABCD with vertices at B(-5, -10), C(-10, 15), D(0, 5), and A(0, 0).
- The dilated quadrilateral A'B'C'D' with coordinates A'(0, 0), B'(1, 2), C'(2, -3), and D'(0, -1).

These steps will yield a graphical representation of both quadrilaterals. If you use graph paper or a graphing software, you can plot these points accordingly for the best visual representation.