Algebraic Dilation About a Point | Geometry (TX TEKS) | Khan Academy

Understanding Dilation About a Point

• 🎯 This video explains how to perform a dilation on a geometric figure about a point that is not the origin.
• 🔑 The process involves three transformations: translating the figure so the center of dilation is at the origin, dilating about the origin, and then translating back.

Step 1: Translation to the Origin

• ➡️ To shift the center of dilation (e.g., point -1, 2) to the origin (0,0), we apply a translation.
• ➕ This translation involves adding 1 to the x-coordinate and subtracting 2 from the y-coordinate of every point in the figure.
• 📉 After this shift, the triangle's vertices are at (6, 3), (9, 0), and (0, -3).

Step 2: Dilation About the Origin

• ✖️ With the figure now centered relative to the origin, we apply the dilation with the given scale factor (2/3).
• 📏 This is done by multiplying each x and y coordinate by the scale factor: (2/3 * x, 2/3 * y).
• 💡 The vertices of the dilated triangle become (4, 2), (6, 0), and (0, -2).

Step 3: Translation Back

• ↩️ The final step is to reverse the initial translation to move the center of dilation back to its original position (-1, 2).
• ➕ This is achieved by subtracting 1 from the x-coordinate and adding 2 to the y-coordinate of each point from the previous step.
• 📝 The final coordinates for the image of the triangle are derived by simplifying the algebraic expressions for the transformations.

Final Algebraic Representation

• 🧮 The overall dilation transformation can be represented algebraically.
• 📈 The x-coordinate of the final image is given by (2/3 * x - 1/3).
• 📊 The y-coordinate of the final image is given by (2/3 * y + 2/3).