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Mathematics › 📐 Grade 8 › Dilations & Similarity: Same Shape, Different Size
📐 Grade 8 · Lesson 10 of 15

Dilations & Similarity: Same Shape, Different Size

A dilation grows or shrinks a figure from a fixed centre — angles stay put, but area scales by the square of the factor.

Grade 8Middle school geometry
Dilations & Similarity: Same Shape, Different Size — illustration
💡
The big idea: A <strong>dilation</strong> resizes a figure from a fixed <strong>centre</strong> by a <strong>scale factor k</strong>: every length is multiplied by k. Angles never change under a dilation, only size does — so a dilation (possibly combined with a rigid motion) always produces a <strong>similar</strong> figure: same shape, corresponding angles equal, corresponding sides proportional. Because a dilation stretches every length by k, it stretches area by <strong>k²</strong> — the same rule Grade 7 met when scaling drawings.
🎯 By the end, you'll be able to
  • Describe how a dilation with a given centre and scale factor k transforms a figure
  • Explain why angles are unchanged but area scales by k² under a dilation
  • Determine whether two figures are similar and use proportional sides to find missing lengths
  • Apply the coordinate rule for a dilation centred at the origin, including for a negative scale factor
📎 You should already know
  • Scale drawings and scale factor (Grade 7)
  • Translations, rotations, reflections & congruence (Grade 8)

Resizing from a fixed point

Unlike a translation, rotation, or reflection, a dilation is not a rigid motion — it deliberately changes size. A dilation is defined by two things: a fixed centre point and a scale factor k. Every point of the figure moves directly toward or away from the centre, landing at a distance k times its original distance from the centre.

Because every point moves by the same factor k, the resized figure keeps exactly the same proportions as the original — just bigger or smaller.

🔑 What changes and what doesn't

Lengths (sides, distances from the centre) are multiplied by k.

Angles are completely unchanged — a dilation never distorts an angle's measure.

Area is multiplied by k², exactly as with the scale drawings you met in Grade 7: since both the length and width of any region scale by k, area scales by k × k.

Similarity: what a dilation produces

A figure formed by dilating another figure — optionally followed by a translation, rotation, or reflection to reposition it — is called similar to the original. Similar figures always satisfy two matched conditions at once:

  • Corresponding angles are equal.
  • Corresponding sides are proportional (all in the same ratio, k).

Both conditions must hold. A figure with equal angles but non-proportional sides is not similar, and neither is one with proportional sides but mismatched angles.

\[ (x,y)\;\xrightarrow{\text{dilate, centre }O,\text{ factor }k}\;(kx,\;ky) \]
A dilation centred at the origin multiplies every coordinate by the scale factor k.
🎮 Dilation Lab LIVE
Drag the scale factor slider and watch every side length scale by k while every angle stays exactly the same — and see area grow by k².

Reading the scale factor

The value of k tells you exactly what happens: when k > 1, the figure enlarges, moving further from the centre. When 0 < k < 1, the figure shrinks, moving closer to the centre. A negative k sends every point straight through the centre to the opposite side, producing a figure that is also rotated 180° about the centre as it resizes.

📝 Worked example: Triangle ABC has sides 6, 8, and 10. It is dilated by scale factor k = 1.5 from a centre outside the triangle. Find the side lengths of the image, and its area if the original area is 24.
  1. Lengths scale by k = 1.5: sides become \( 6\times1.5=9,\ 8\times1.5=12,\ 10\times1.5=15 \).
  2. Area scales by k² = 1.5² = 2.25.
  3. New area \( = 24 \times 2.25 = 54 \).
✓ Image sides: <strong>9, 12, 15</strong>; image area: <strong>54</strong>. The angles of the image are identical to the original 90°/&hellip; angles — only lengths changed.
📝 Worked example: A point (2, −3) is dilated from the origin with scale factor k = −2. Find the image point.
  1. Apply the rule \( (x,y)\to(kx,ky) \) with k = −2.
  2. New x: \( 2\times(-2) = -4 \). New y: \( -3\times(-2) = 6 \).
  3. Because k is negative, the image lands on the opposite side of the centre from the original point, in addition to being twice as far away.
✓ The image point is <strong>(−4, 6)</strong>.
⚠️ Area scales by k² — and negative k flips through the centre

Grade 7's scale-drawings lesson already established that area scales by k², never by k itself — a dilation with k = 3 makes lengths 3× longer but area 9× larger, not 3×. That rule carries over unchanged here.

Second trap: for 0 < k < 1 the figure shrinks (many students expect any dilation to enlarge). And for negative k, every point passes straight through the centre to the opposite side — the image is not just resized, it appears on the far side of the centre point.

✨ Congruent vs. similar — don't mix them up
Rigid motions (translate, rotate, reflect) change position only and produce a congruent figure — same size, same shape. A dilation changes size and produces a similar figure — same shape, but not necessarily the same size (unless k = 1). Every congruent figure is also similar (with k = 1), but a similar figure is only congruent when its scale factor happens to be exactly 1 or −1.

Check your understanding

1. A dilation has scale factor k = 4. How does it affect the angles of the figure?
A dilation never changes angle measures — it only rescales lengths. Angles stay exactly the same.
2. A square with area 10 cm² is dilated by scale factor k = 3. What is the area of the image?
Area scales by k² = 9, so 10 × 9 = 90 cm² — not 10 × 3 = 30 cm², which mistakes area scaling for length scaling.
3. Point (5, 2) is dilated from the origin with scale factor k = 0.5. What is the image?
The dilation rule (x, y) → (kx, ky) with k = 0.5 gives (5 × 0.5, 2 × 0.5) = (2.5, 1).
4. Which pair of conditions together define similar figures?
Similarity requires both conditions at once: matching angles and sides that are all in the same ratio.
5. A point is dilated from the origin with scale factor k = −1. What happens to it?
A negative scale factor sends the point through the centre to the opposite side. With |k| = 1, the distance from the centre is unchanged, only the direction flips — this is the same as a 180° rotation about the centre.
✅ Key takeaways
  • A dilation resizes a figure from a fixed centre by scale factor k: every length is multiplied by k.
  • Angles are unchanged by a dilation; only size changes — the image is similar, not congruent, to the original (unless k = 1).
  • Area scales by k², the same rule established for scale drawings in Grade 7 — not by k itself.
  • k > 1 enlarges, 0 < k < 1 shrinks, and a negative k sends points through the centre to the opposite side.
  • Similar figures require both corresponding angles equal AND corresponding sides proportional; rigid motions give congruence, dilations give similarity.