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Describing Types of Dilations
There are distinct differences between an enlargement and a reduction with dilations. Describe the key facts about both types of dilations.
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Short answer: a dilation is a similarity transformation about a center O that multiplies every vector OP by a scale factor k. If |k|>1 it is an enlargement (image bigger), if 0<|k|<1 it is a reduction (image smaller). Key facts:
General facts for all dilations
- Center: the center O is fixed; every point P maps to P′ on line OP (collinear with O and P).
- Position formula: P′ = O + k(P − O) (in coordinates: if O = (x0,y0) and P = (x,y), then P′ = (x0 + k(x−x0), y0 + k(y−y0))).
- Distances from the center scale by |k|: OP′ = |k|·OP.
- Corresponding side lengths scale by |k|; corresponding angles are equal (dilations preserve angle measure).
- Figures are similar: all corresponding linear measures multiply by |k|, area scales by k^2, volume by k^3.
- Orientation: preserved when k>0, reversed when k<0.
- Special values: k = 1 is the identity; k = 0 collapses everything to the center; k = −1 is a half-turn (180° rotation) about the center.
Enlargement (|k| > 1, usually k>1)
- Image is larger than the original: every distance from the center increases by factor k.
- Side ratios: image side / original side = k (>1).
- Area grows by factor k^2 (so area increases).
- If k>1 orientation is preserved; if k<−1 orientation is reversed and the figure is also inverted through the center.
Reduction (0 < |k| < 1, usually 0<k<1)
- Image is smaller: every distance from the center decreases by factor k.
- Side ratios: image side / original side = k (a positive number less than 1).
- Area shrinks by factor k^2 (so area decreases).
- Orientation preserved for positive k; if k is negative (−1<k<0) you get a reduction plus a half-turn (orientation reversed).
Examples
- If k = 2: lengths double, area quadruples.
- If k = 1/3: lengths become one third, area becomes 1/9.
- If k = −2: lengths double but the image is rotated 180° about the center (orientation reversed).
These are the main differences and properties distinguishing enlargements and reductions under dilation.
General facts for all dilations
- Center: the center O is fixed; every point P maps to P′ on line OP (collinear with O and P).
- Position formula: P′ = O + k(P − O) (in coordinates: if O = (x0,y0) and P = (x,y), then P′ = (x0 + k(x−x0), y0 + k(y−y0))).
- Distances from the center scale by |k|: OP′ = |k|·OP.
- Corresponding side lengths scale by |k|; corresponding angles are equal (dilations preserve angle measure).
- Figures are similar: all corresponding linear measures multiply by |k|, area scales by k^2, volume by k^3.
- Orientation: preserved when k>0, reversed when k<0.
- Special values: k = 1 is the identity; k = 0 collapses everything to the center; k = −1 is a half-turn (180° rotation) about the center.
Enlargement (|k| > 1, usually k>1)
- Image is larger than the original: every distance from the center increases by factor k.
- Side ratios: image side / original side = k (>1).
- Area grows by factor k^2 (so area increases).
- If k>1 orientation is preserved; if k<−1 orientation is reversed and the figure is also inverted through the center.
Reduction (0 < |k| < 1, usually 0<k<1)
- Image is smaller: every distance from the center decreases by factor k.
- Side ratios: image side / original side = k (a positive number less than 1).
- Area shrinks by factor k^2 (so area decreases).
- Orientation preserved for positive k; if k is negative (−1<k<0) you get a reduction plus a half-turn (orientation reversed).
Examples
- If k = 2: lengths double, area quadruples.
- If k = 1/3: lengths become one third, area becomes 1/9.
- If k = −2: lengths double but the image is rotated 180° about the center (orientation reversed).
These are the main differences and properties distinguishing enlargements and reductions under dilation.
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