{
  "id": "dilation",
  "version": "d15588678e69",
  "status": "published",
  "name": "Dilation Calculator",
  "question": "Where do points go in a dilation?",
  "summary": "Dilates points in the coordinate plane by a scale factor k about any centre, P′ = C + k(P − C), and says whether the figure is enlarged or reduced, with the length and area factors.",
  "category": "math",
  "subcategory": "geometry",
  "url": "https://www.acalculator.org/math/dilation-calculator",
  "markdown": "https://www.acalculator.org/math/dilation-calculator.md",
  "kind": "function",
  "method": "P′ = C + k(P − C): x′ = cx + k(x − cx), y′ = cy + k(y − cy); lengths × |k|, areas × k².",
  "assumptions": [
    "Typed decimals are read exactly, so each image coordinate is the exact decimal, rounded once to 10 significant figures.",
    "A negative scale factor puts the image on the other side of the centre, the same as a dilation by |k| and a half turn."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "k": {
        "title": "Scale factor k",
        "description": "How many times farther from the centre each image point is. A negative k also turns the figure through 180°.",
        "type": "number",
        "minimum": -1000000,
        "maximum": 1000000
      },
      "cx": {
        "title": "Centre x",
        "description": "The x coordinate of the centre of dilation; 0 when left empty.",
        "type": "number",
        "minimum": -1000000000000,
        "maximum": 1000000000000
      },
      "cy": {
        "title": "Centre y",
        "description": "The y coordinate of the centre of dilation; 0 when left empty.",
        "type": "number",
        "minimum": -1000000000000,
        "maximum": 1000000000000
      },
      "points": {
        "title": "Points",
        "description": "The points of the figure to dilate, A, B, C and so on, up to 12.",
        "type": "array",
        "items": {
          "type": "object",
          "properties": {
            "x": {
              "title": "x",
              "description": "The x coordinate of the point.",
              "type": "number",
              "minimum": -1000000000000,
              "maximum": 1000000000000
            },
            "y": {
              "title": "y",
              "description": "The y coordinate of the point.",
              "type": "number",
              "minimum": -1000000000000,
              "maximum": 1000000000000
            }
          }
        }
      }
    }
  },
  "outputs": {
    "image": {
      "label": "Image points",
      "description": "Each point after the dilation, as A′(x, y); B′(x, y) and so on.",
      "format": "text"
    },
    "kind": {
      "label": "Type of dilation",
      "description": "An enlargement when |k| > 1, a reduction when |k| < 1, the same size when |k| = 1.",
      "format": "text"
    },
    "lengthFactor": {
      "label": "Length factor",
      "description": "|k|: every length of the image is this many times the original length.",
      "format": "number"
    },
    "areaFactor": {
      "label": "Area factor",
      "description": "k²: the area of the image is this many times the original area.",
      "format": "number"
    },
    "steps": {
      "label": "Working",
      "description": "For each point, x′ = cx + k(x − cx) and y′ = cy + k(y − cy).",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "k": 2,
      "cx": 0,
      "cy": 0,
      "points": [
        {
          "x": -4,
          "y": 6
        },
        {
          "x": 2,
          "y": 1
        },
        {
          "x": 5,
          "y": 4
        }
      ]
    },
    "outputs": {
      "image": "A′(−8, 12); B′(4, 2); C′(10, 8)",
      "kind": "Enlargement",
      "lengthFactor": 2,
      "areaFactor": 4,
      "steps": "centre (0, 0), k = 2; A(−4, 6) → (0 + 2 × (−4 − 0), 0 + 2 × (6 − 0)) = A′(−8, 12); B(2, 1) → (0 + 2 × (2 − 0), 0 + 2 × (1 − 0)) = B′(4, 2); C(5, 4) → (0 + 2 × (5 − 0), 0 + 2 × (4 − 0)) = C′(10, 8)"
    },
    "text": "The image points are A′(−8, 12); B′(4, 2); C′(10, 8)."
  },
  "examples": [
    {
      "given": {
        "k": 2,
        "cx": 0,
        "cy": 0,
        "points": [
          {
            "x": -4,
            "y": 6
          }
        ]
      },
      "expect": {
        "image": "A′(−8, 12)",
        "kind": "Enlargement",
        "areaFactor": 4
      },
      "source": "CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the rule (x, y) → (kx, ky) about the origin; A(−4, 6) with k = 2 gives A′(−8, 12); A(9, −13) with k = 1/2 gives A′(4.5, −6.5)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02)"
    },
    {
      "given": {
        "k": 0.5,
        "cx": 0,
        "cy": 0,
        "points": [
          {
            "x": 9,
            "y": -13
          }
        ]
      },
      "expect": {
        "image": "A′(4.5, −6.5)",
        "kind": "Reduction",
        "lengthFactor": 0.5,
        "areaFactor": 0.25
      },
      "source": "CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the rule (x, y) → (kx, ky) about the origin; A(−4, 6) with k = 2 gives A′(−8, 12); A(9, −13) with k = 1/2 gives A′(4.5, −6.5)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02)"
    },
    {
      "given": {
        "k": 3,
        "cx": 1,
        "cy": 2,
        "points": [
          {
            "x": 2,
            "y": 3
          },
          {
            "x": 4,
            "y": 2
          },
          {
            "x": 1,
            "y": 5
          }
        ]
      },
      "expect": {
        "image": "A′(4, 5); B′(10, 2); C′(1, 11)",
        "areaFactor": 9
      },
      "source": "hand calculation in content.mdx: x′ = 1 + 3(x − 1), y′ = 2 + 3(y − 2); OpenStax, Elementary Algebra 2e, §8.7 Solve Proportion and Similar Figure Applications (similar figures have corresponding sides in the same ratio), https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications (retrieved 2026-10-02)"
    },
    {
      "given": {
        "k": -0.5,
        "cx": 0,
        "cy": 0,
        "points": [
          {
            "x": 4,
            "y": -2
          }
        ]
      },
      "expect": {
        "image": "A′(−2, 1)",
        "kind": "Reduction, turned 180° about the centre",
        "areaFactor": 0.25
      },
      "source": "hand calculation in content.mdx: (−0.5 × 4, −0.5 × −2); CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the rule (x, y) → (kx, ky) about the origin; A(−4, 6) with k = 2 gives A′(−8, 12); A(9, −13) with k = 1/2 gives A′(4.5, −6.5)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02)"
    }
  ],
  "sources": [
    "CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the mapping (x, y) → (kx, ky); worked examples with k = 2 and k = ½), on K12 LibreTexts. https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02)",
    "OpenStax, Elementary Algebra 2e, §8.7 Solve Proportion and Similar Figure Applications (similar figures have corresponding sides in the same ratio), CC BY 4.0. https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications (retrieved 2026-10-02)"
  ],
  "related": [
    "scale-factor",
    "midpoint",
    "distance-formula",
    "slope"
  ],
  "changelog": []
}