Examples of Dilations with Different Scale Factors

📚 Quick Study Guide

• 📏 Dilation Definition: A dilation is a transformation that changes the size of a figure. It requires a center point and a scale factor.
• 🔢 Scale Factor (k): The scale factor determines how much larger or smaller the image will be.
• 📈 Enlargement (k > 1): If the scale factor is greater than 1, the image is larger than the original.
• 📉 Reduction (0 < k < 1): If the scale factor is between 0 and 1, the image is smaller than the original.
• 📍 Center of Dilation: The point about which the figure is enlarged or reduced.
• 📐 Coordinate Rule: When dilating a point $(x, y)$ by a scale factor $k$ centered at the origin, the new coordinates are $(kx, ky)$.
• ✨ Similarity: Dilations produce similar figures. This means the angles stay the same, but the side lengths change proportionally.

Practice Quiz

1. A triangle has vertices A(1, 2), B(3, 4), and C(5, 2). It is dilated by a scale factor of 2 centered at the origin. What are the coordinates of the dilated triangle's vertices?

   1. A'(2, 4), B'(6, 8), C'(10, 4)
   2. A'(0.5, 1), B'(1.5, 2), C'(2.5, 1)
   3. A'(3, 4), B'(5, 6), C'(7, 4)
   4. A'(1, 4), B'(3, 8), C'(5, 4)

2. A square with side length 4 is dilated by a scale factor of 0.5. What is the side length of the dilated square?

   1. 8
   2. 2
   3. 16
   4. 0.5

3. If a point (6, -3) is dilated by a scale factor of $\frac{1}{3}$, what are the coordinates of the image?

   1. (2, -1)
   2. (18, -9)
   3. (6$\frac{1}{3}$, -3$\frac{1}{3}$)
   4. (3, -1.5)

4. A line segment has endpoints P(-2, 2) and Q(4, -2). If it is dilated by a scale factor of 3 centered at the origin, what are the coordinates of P' and Q'?

   1. P'(-6, 6), Q'(12, -6)
   2. P'(-$\frac{2}{3}$, $\frac{2}{3}$), Q'($\frac{4}{3}$, -$\frac{2}{3}$)
   3. P'(1, -1), Q'(-2, 1)
   4. P'(0, 0), Q'(7, -7)

5. A circle with radius 5 is dilated by a scale factor of 1.5. What is the radius of the new circle?

   1. 3.5
   2. 7.5
   3. 6.5
   4. 5

6. Triangle XYZ has vertices X(0,0), Y(2,0), and Z(1,3). If it's dilated by a scale factor of 4 centered at the origin, what are the new coordinates of Z'?

   1. (4, 12)
   2. (0.25, 0.75)
   3. (5, 7)
   4. (1, 3)

7. A rectangle with vertices (1,1), (1,3), (4,1), and (4,3) is dilated by a scale factor of 0.25 centered at the origin. What are the coordinates of the dilated rectangle's vertices?

   1. (0.25, 0.25), (0.25, 0.75), (1, 0.25), (1, 0.75)
   2. (4,4), (4,12), (16,4), (16,12)
   3. (0.75, 0.75), (0.75, 1.25), (1.25, 0.75), (1.25, 1.25)
   4. (2,2), (2,6), (8,2), (8,6)