Dilations On The Coordinate Plane Answer Key

Introduction

Understanding dilations on the coordinate plane is essential for mastering transformations in geometry. An answer key for these problems typically includes the steps needed to identify the scale factor, apply the dilation formula, and verify the resulting coordinates. This article provides a clear, step‑by‑step guide, a comprehensive answer key, and explanations that help students of all backgrounds solve dilation problems confidently.

What Is a Dilation?

A dilation is a transformation that produces an image of the same shape as the original but changes its size. The key element is the scale factor (k), a number that tells how much the figure is enlarged (k > 1) or reduced (0 < k < 1). When performed on a coordinate plane, each point ((x, y)) is mapped to a new point ((kx, ky)).

Key points to remember

• Scale factor determines the direction and magnitude of the change.
• The origin ((0,0)) is the center of dilation unless another point is specified.
• Distances from the center are multiplied by the scale factor, preserving the shape’s proportions.

Steps to Perform a Dilation on the Coordinate Plane

1. Identify the center of dilation
   • Most problems use the origin, but the center can be any point ((h, k)).
2. Determine the scale factor
   • Write it as a fraction or decimal; for reduction, use a value between 0 and 1.
3. Apply the dilation formula
   • For each original point ((x, y)), calculate the image point:
     [ (x', y') = (h + k(x - h),; k + k(y - k)) ]
   • If the center is the origin, the formula simplifies to ((x', y') = (kx,; ky)).
4. Plot the transformed points
   • Connect the new points in the same order as the original figure to visualize the dilation.
5. Verify the results
   • Check that the distances between corresponding points are in the ratio of the scale factor.

Answer Key for Common Dilation Problems

Below is a complete answer key for typical dilation exercises found in textbooks and worksheets. Each problem includes the given information, the solution steps, and the final coordinates.

Problem 1

Given: Triangle with vertices (A(1, 2)), (B(4, 5)), (C(2, 6)). Dilate with a scale factor of 3 about the origin Not complicated — just consistent..

Solution:

• Scale factor (k = 3).
• Apply ((x', y') = (3x, 3y)) to each vertex:
  • (A': (3·1, 3·2) = (3, 6))
  • (B': (3·4, 3·5) = (12, 15))
  • (C': (3·2, 3·6) = (6, 18))

Answer Key: (A'(3, 6), B'(12, 15), C'(6, 18)).

Problem 2

Given: Square with vertices (P(0, 0)), (Q(2, 0)), (R(2, 2)), (S(0, 2)). Dilate with a scale factor of (\frac{1}{2}) about the point ((1, 1)).

Solution:

• Center ((h, k) = (1, 1)), scale factor (k = \frac{1}{2}).
• Use the full formula:
  [ x' = h + k(x - h), \quad y' = k + k(y - k) ]
• Compute each vertex:
  • (P(0,0): x' = 1 + \frac{1}{2}(0 - 1) = 1 - 0.5 = 0.5,; y' = 1 + \frac{1}{2}(0 - 1) = 0.5) → (P'(0.5, 0.5))
  • (Q(2,0): x' = 1 + \frac{1}{2}(2 - 1) = 1 + 0.5 = 1.5,; y' = 1 + \frac{1}{2}(0 - 1) = 0.5) → (Q'(1.5, 0.5))
  • (R(2,2): x' = 1 + \frac{1}{2}(2 - 1) = 1.5,; y' = 1 + \frac{1}{2}(2 - 1) = 1.5) → (R'(1.5, 1.5))
  • (S(0,2): x' = 1 + \frac{1}{2}(0 - 1) = 0.5,; y' = 1 + \frac{1}{2}(2 - 1) = 1.5) → (S'(0.5, 1.5))

Answer Key: (P'(0.5, 0.5), Q'(1.5, 0.5), R'(1.5, 1.5), S'(0.5, 1.5)) Took long enough..

Problem 3

Given: Rectangle with vertices (M(-3, 1)), (N(-3, 4)), (O(2, 4)), (P(2, 1)). Dilate with a scale factor of (-2) about the origin.

Solution:

• Scale factor (k = -2) (negative indicates a dilation combined with a rotation of 180°).
• Apply ((x', y') = (-2x, -2y)):
  • (M': (-2·-3, -2·1) = (6, -2))
  • (N': (-2·-3, -2·4) = (6, -8))
  • (O': (-2·2, -2·4) = (-4, -8))
  • (P': (-2·2, -2·1) = (-4, -2))

Answer Key: (M'(6, -2), N'(6, -8), O'(-4, -8), P'(-4, -2)).

Problem 4

Given: Point (A(5, -1)) is dilated to (A'(10, -5)). Find the scale factor and state the center of dilation That's the part that actually makes a difference..

Solution:

• Compare coordinates: (10 = k·5) → (k = 2).
• Also (-5 = k·(-1)) → (k = 5) (contradiction).
• Since the ratios differ, the center is not the origin.