How to Perform a Dilation on a Coordinate Plane: Step-by-Step

📚 Understanding Dilations

A dilation is a transformation that changes the size of a figure. It either enlarges (stretches) or reduces (shrinks) the figure. The amount of enlargement or reduction is determined by the scale factor.

• 📏Scale Factor: The scale factor, often denoted as $k$, determines how much larger or smaller the image will be compared to the original.
• 📍Center of Dilation: The center of dilation is a fixed point in the plane about which all points are expanded or contracted.

📝 Objectives

• 🎯 Understand the concept of dilation.
• 🔢 Perform dilations on a coordinate plane.
• 📈 Determine the coordinates of the image after dilation.

🧮 Materials

• graph paper
• ✏️ pencil
• 📏 ruler
• 🧠 calculator (optional)

Warm-up (5 mins)

Review coordinate plane basics.

• 📍 Plot the following points on a coordinate plane: A(2, 3), B(-1, 4), C(0, -2).
• ➕ Identify the coordinates of the point located 3 units to the right and 2 units down from the origin. Answer: (3, -2)

Main Instruction

Step 1: Identify the Center of Dilation and Scale Factor

• 🔍 Usually, the center of dilation is the origin (0, 0). If it's different, adjustments are needed.
• 🔢 The scale factor ($k$) will be given in the problem.

Step 2: Multiply Coordinates by the Scale Factor

• ✖️ If the center of dilation is the origin, multiply each coordinate of the original figure by the scale factor ($k$).
• 📐 For a point (x, y), the new coordinates after dilation will be (kx, ky).

Step 3: Plot the New Coordinates

• 📈 Plot the new coordinates on the coordinate plane.
• ✏️ Connect the points to form the dilated image.

Example

Dilate triangle ABC with vertices A(1, 1), B(2, 1), and C(1, 2) by a scale factor of 2, with the center of dilation at the origin.

• 📍 A'(2*1, 2*1) = A'(2, 2)
• 📐 B'(2*2, 2*1) = B'(4, 2)
• 📌 C'(2*1, 2*2) = C'(2, 4)

Plot A'(2, 2), B'(4, 2), and C'(2, 4) and connect the points.

Assessment

Dilate the rectangle with vertices P(0, 0), Q(0, 2), R(3, 2), and S(3, 0) by a scale factor of 0.5, with the center of dilation at the origin. What are the coordinates of the dilated rectangle?

• 📍 P'(0*0.5, 0*0.5) = P'(0, 0)
• 📐 Q'(0*0.5, 2*0.5) = Q'(0, 1)
• 📌 R'(3*0.5, 2*0.5) = R'(1.5, 1)
• 💡 S'(3*0.5, 0*0.5) = S'(1.5, 0)

Answer: P'(0, 0), Q'(0, 1), R'(1.5, 1), S'(1.5, 0)

Practice Quiz

Here are some extra practice questions to solidify your understanding:

1. 📌 Dilate the point (3, -2) by a scale factor of 3.
2. 📐 Dilate the line segment with endpoints (1, 4) and (5, 2) by a scale factor of 0.5.
3. 📍 Dilate the triangle with vertices (0, 0), (2, 0), and (0, 3) by a scale factor of 2.5.