Question 1174947: Show that if ∆A'B'C' is the image of ∆ABC under a dilation with center O and scale
factor k, then
∆A'B'C' ∆ABC = k
2
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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).
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