What are dilations when the center is not the origin?

📚 Understanding Dilations When the Center is Not the Origin

Dilation is a transformation that changes the size of a figure. When the center of dilation is the origin (0,0), things are pretty straightforward. But when it's not, we need to take a few extra steps.

📜 A Little History

The concept of dilation has been around for centuries, rooted in geometry and perspective drawing. Early mathematicians and artists recognized the need to represent objects at different scales, leading to the formalization of dilation as a geometric transformation.

📌 Key Principles

• 📏 The Scale Factor: The scale factor ($k$) determines how much larger or smaller the image will be. If $k > 1$, the image is larger; if $0 < k < 1$, the image is smaller.
• 📍 The Center of Dilation: This is the fixed point from which all points are scaled. When it's not the origin, it requires a shift.
• ⚙️ Translation: We use translation to shift the figure so that the center of dilation is at the origin, perform the dilation, and then shift it back.

🧭 Step-by-Step Guide

1. Translate: Move the figure so that the center of dilation coincides with the origin. If the center of dilation is $(h, k)$, translate the figure by $(-h, -k)$.
2. Dilate: Apply the dilation with the scale factor $k$. Multiply the coordinates of each point by $k$: $(x, y) \rightarrow (kx, ky)$.
3. Translate Back: Move the figure back to its original position by translating by $(h, k)$.

✏️ Formula Breakdown

If a point $(x, y)$ is dilated with center $(h, k)$ and scale factor $k$, the new point $(x', y')$ is given by:

$x' = h + k(x - h)$

$y' = k + k(y - k)$

➕ Example Problem

Let's dilate the point $A(2, 3)$ with center $C(1, 1)$ and scale factor $2$.

1. Translate: Subtract $(1, 1)$ from $(2, 3)$ to get $(1, 2)$.
2. Dilate: Multiply $(1, 2)$ by $2$ to get $(2, 4)$.
3. Translate Back: Add $(1, 1)$ to $(2, 4)$ to get $(3, 5)$.

So, the dilated point $A'$ is $(3, 5)$.

💡 Real-World Examples

• 🗺️ Mapmaking: Creating maps involves scaling down real-world distances, using a center point for reference.
• 📸 Photography: Enlarging or reducing photos uses dilation principles, with the lens acting as the center.
• 📐 Architecture: Blueprints are scaled-down versions of actual buildings, using a specific scale factor.

📝 Practice Quiz

1. Dilate the point $(4, 6)$ with center $(2, 2)$ and scale factor $3$.
2. A triangle has vertices $A(1, 2)$, $B(3, 4)$, and $C(5, 2)$. Dilate the triangle with center $(0, 0)$ and scale factor $0.5$.
3. Dilate the line segment connecting $(2, 3)$ and $(4, 5)$ with center $(1, 1)$ and scale factor $2$.
4. A square has vertices $(1, 1)$, $(1, 3)$, $(3, 3)$, and $(3, 1)$. Dilate the square with center $(2, 2)$ and scale factor $1.5$.
5. Dilate the circle with center $(3, 2)$ and radius $1$ with center $(1, 1)$ and scale factor $2$.

🔑 Conclusion

Dilation with a center other than the origin involves a combination of translation and dilation. By understanding these steps, you can accurately perform dilations in any situation. Keep practicing, and you'll master it in no time!