IG IDEAS MATH pcole057 \#19 Check Skills Review Save / E Listen Use the given vertices to graph quadrilateral $$ R S T U $$ and its image after a dilation centered at the origin with scale factor $$ k=-4 $$. $$ R(-7,-1), S(2,5), T(-2,-3) ; U(-3,-3) $$ Polygon Undo Redo Reset

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Certainly! Let's walk through the process of graphing quadrilateral $$ RSTU $$ and its image after a dilation centered at the origin with a scale factor of $$ k = -4 $$.

## **1. Understanding Dilation**

- **Dilation** is a transformation that resizes a figure by a scaling factor ($$ k $$) while maintaining its shape. 
- The **center of dilation** is the fixed point from which all points are expanded or contracted.
- A **scale factor** ($$ k = -4 $$) indicates:
  - **Magnitude (4)**: The figure will be enlarged 4 times its original size.
  - **Negative Sign (-)**: The figure will also be **reflected** across the center of dilation (in this case, the origin).

## **2. Original Quadrilateral $$ RSTU $$**

Given vertices:
- $$ R(-7, -1) $$
- $$ S(2, 5) $$
- $$ T(-2, -3) $$
- $$ U(-3, -3) $$

### **Plotting $$ RSTU $$:**

1. **Plot each vertex** on the Cartesian plane.
2. **Connect the vertices** in order to form quadrilateral $$ RSTU $$.

*Since I can’t provide a visual graph here, you can plot these points on graph paper or using a graphing tool.*

## **3. Applying the Dilation**

**Formula for Dilation:**
$$
(x, y) \rightarrow (k \cdot x, k \cdot y)
$$
Given $$ k = -4 $$, we'll apply this to each vertex.

### **Calculations:**

1. **Point $$ R(-7, -1) $$:**
   $$
   R' = (-4) \times (-7, -1) = (28, 4)
   $$
   
2. **Point $$ S(2, 5) $$:**
   $$
   S' = (-4) \times (2, 5) = (-8, -20)
   $$
   
3. **Point $$ T(-2, -3) $$:**
   $$
   T' = (-4) \times (-2, -3) = (8, 12)
   $$
   
4. **Point $$ U(-3, -3) $$:**
   $$
   U' = (-4) \times (-3, -3) = (12, 12)
   $$

### **Dilated Quadrilateral $$ R'S'T'U' $$:**

- $$ R'(28, 4) $$
- $$ S'(-8, -20) $$
- $$ T'(8, 12) $$
- $$ U'(12, 12) $$

## **4. Plotting the Dilated Quadrilateral $$ R'S'T'U' $$**

1. **Plot each dilated vertex** on the same Cartesian plane.
2. **Connect the dilated vertices** in order to form quadrilateral $$ R'S'T'U' $$.

**Key Observations:**
- The negative scale factor reflects each point through the origin.
- The figure is enlarged by a factor of 4.
- The orientation of the dilated quadrilateral is opposite to the original due to the negative scale factor.

## **5. Summary of Coordinates**

| Original Point | Coordinates | Dilated Point $$ (k = -4) $$ | Coordinates  |
|----------------|-------------|-------------------------------|--------------|
| R              | (-7, -1)    | $$ R' = -4 \times R $$        | (28, 4)      |
| S              | (2, 5)      | $$ S' = -4 \times S $$        | (-8, -20)    |
| T              | (-2, -3)    | $$ T' = -4 \times T $$        | (8, 12)      |
| U              | (-3, -3)    | $$ U' = -4 \times U $$        | (12, 12)     |

## **6. Graphing Tips**

- **Use graph paper or a digital graphing tool** for accuracy.
- **Label both quadrilaterals** clearly to distinguish between the original and the dilated image.
- **Check reflection:** Since $$ k $$ is negative, ensure that the dilated quadrilateral is a mirror image across the origin.

## **7. Final Notes**

- Always apply the dilation formula to each vertex.
- Remember that a negative scale factor not only scales but also reflects the figure.
- The center of dilation (origin) remains unchanged during dilation.

If you follow these steps, you'll accurately graph both the original quadrilateral $$ RSTU $$ and its dilated image $$ R'S'T'U' $$.
To graph quadrilateral $$ RSTU $$ and its image after a dilation centered at the origin with a scale factor of $$ k = -4 $$, follow these steps: 1. **Plot the original vertices**: - $$ R(-7, -1) $$ - $$ S(2, 5) $$ - $$ T(-2, -3) $$ - $$ U(-3, -3) $$ 2. **Apply the dilation formula** to each vertex: - $$ R' = (-4) \times (-7, -1) = (28, 4) $$ - $$ S' = (-4) \times (2, 5) = (-8, -20) $$ - $$ T' = (-4) \times (-2, -3) = (8, 12) $$ - $$ U' = (-4) \times (-3, -3) = (12, 12) $$ 3. **Plot the dilated vertices**: - $$ R'(28, 4) $$ - $$ S'(-8, -20) $$ - $$ T'(8, 12) $$ - $$ U'(12, 12) $$ 4. **Connect the vertices** to form the original and dilated quadrilaterals. **Final Quadrilaterals:** - Original: $$ R(-7, -1) $$, $$ S(2, 5) $$, $$ T(-2, -3) $$, $$ U(-3, -3) $$ - Dilated: $$ R'(28, 4) $$, $$ S'(-8, -20) $$, $$ T'(8, 12) $$, $$ U'(12, 12) $$ Ensure to plot these points accurately on a graph to visualize the transformation.

IG IDEAS MATH pcole057 \#19 Check Skills Review Save / E Listen Use the given vertices to graph quadrilateral \( R S T U \) and its image after a dilation centered at the origin with scale factor \( k=-4 \). \[ R(-7,-1), S(2,5), T(-2,-3) ; U(-3,-3) \] Polygon Undo Redo Reset

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