Steps to Dilate a Figure from a Point Not at the Origin

📚 Understanding Dilation: A Comprehensive Guide

Dilation is a transformation that changes the size of a figure. It either enlarges (stretches) or reduces (shrinks) the figure, but it doesn't change its shape. A key element of dilation is the center of dilation, which is a fixed point that determines how the figure changes.

📜 Historical Context

The concept of dilation has been used in geometry for centuries, underpinning principles used in perspective drawing and mapmaking. Early mathematicians explored similar figures and proportionality, laying the groundwork for the formal definition of dilation we use today.

📌 Key Principles of Dilation

• 📏 Scale Factor: The scale factor, often denoted as 'k', determines the amount of enlargement or reduction. If $k > 1$, the figure is enlarged. If $0 < k < 1$, the figure is reduced. If $k = 1$, the size stays the same.
• 🎯 Center of Dilation: This is the fixed point from which the dilation is performed. All points of the original figure move away from or towards this point, maintaining their relative positions.
• ↔️ Similarity: Dilation produces similar figures. This means that the corresponding angles of the original and dilated figures are equal, and the corresponding sides are proportional.

✍️ Steps to Dilate a Figure from a Point Not at the Origin

When the center of dilation is not at the origin, we need to follow a few extra steps to accurately dilate the figure. Here's a breakdown:

• 1️⃣ Translate the Center to the Origin: First, translate the entire figure (including the center of dilation) so that the center of dilation is moved to the origin (0,0). If the original center is at point (a, b), subtract 'a' from all x-coordinates and 'b' from all y-coordinates of the figure's vertices and the center of dilation.
• 🔢 Apply the Dilation: Multiply the coordinates of each vertex of the translated figure by the scale factor 'k'. So, $(x, y)$ becomes $(kx, ky)$.
• ↩️ Translate Back: Finally, translate the figure back to its original position by adding 'a' to all x-coordinates and 'b' to all y-coordinates. This reverses the initial translation.

💡 Example

Let's say we have triangle ABC with vertices A(1, 2), B(3, 4), and C(5, 2). We want to dilate it by a scale factor of 2 with a center of dilation at D(2, 3).

1. 🌍 Translate: Subtract (2, 3) from all points:
   A'(1-2, 2-3) = A'(-1, -1)
   B'(3-2, 4-3) = B'(1, 1)
   C'(5-2, 2-3) = C'(3, -1)
   D'(2-2, 3-3) = D'(0, 0)
2. 📈 Dilate: Multiply by the scale factor (k=2):
   A''(-1*2, -1*2) = A''(-2, -2)
   B''(1*2, 1*2) = B''(2, 2)
   C''(3*2, -1*2) = C''(6, -2)
3. 📍 Translate Back: Add (2, 3) to all points:
   A'''(-2+2, -2+3) = A'''(0, 1)
   B'''(2+2, 2+3) = B'''(4, 5)
   C'''(6+2, -2+3) = C'''(8, 1)

So, the vertices of the dilated triangle are A'''(0, 1), B'''(4, 5), and C'''(8, 1).

🧪 Real-World Applications

• 🗺️ Mapmaking: Cartographers use dilation to create maps at different scales. A map is essentially a dilation of the real world onto a smaller surface.
• 📸 Photography: When you zoom in or out on a photograph, you are essentially dilating the image.
• 📐 Engineering and Architecture: Engineers and architects use dilation to create scaled models of structures, ensuring that proportions are maintained.

✅ Conclusion

Dilating figures from a point not at the origin involves a few straightforward steps: translating, dilating, and translating back. By understanding these steps and practicing with examples, you can master this important geometric transformation.