Question 1174947
You've got a slight error in your question. What you're likely aiming to show is that the **ratio of corresponding sides** of the triangles is equal to the scale factor k.

Here's the proof:

**Understanding Dilation**

* A dilation is a transformation that changes the size of a figure but not its shape.
* The center of dilation (O) is a fixed point.
* The scale factor (k) determines how much the figure is enlarged or reduced.
* If k > 1, the figure is enlarged.
* If 0 < k < 1, the figure is reduced.
* If k < 0, the figure is dilated and reflected through the center of dilation.

**Proof**

1.  **Definitions:**
    * ∆ABC is the original triangle.
    * ∆A'B'C' is the image of ∆ABC after a dilation with center O and scale factor k.
    * This means:
        * OA' = k * OA
        * OB' = k * OB
        * OC' = k * OC

2.  **Consider Sides AB and A'B'**
    * We want to show that A'B' / AB = k.

3.  **Vector Representation:**
    * We can represent the sides as vectors:
        * AB = OB - OA
        * A'B' = OB' - OA'

4.  **Substitute Dilation Relationships:**
    * A'B' = (k * OB) - (k * OA)
    * A'B' = k(OB - OA)
    * A'B' = k * AB

5.  **Ratio of Sides:**
    * A'B' / AB = (k * AB) / AB
    * A'B' / AB = k

6.  **Similar Proof for Other Sides:**
    * By the same logic, we can show that:
        * B'C' / BC = k
        * A'C' / AC = k

**Conclusion**

Therefore, if ∆A'B'C' is the image of ∆ABC under a dilation with center O and scale factor k, then the ratio of corresponding sides is equal to k.

**In simpler terms:**

The lengths of all the sides of the new triangle (∆A'B'C') are exactly k times the lengths of the corresponding sides of the original triangle (∆ABC).