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Dilations and scale drawings

Geometry · level 5, lesson 4 of 6

A question to start

Olympus Mons on Mars is the tallest volcano known in the solar system: it rises about 22 km above the plains around it, more than twice the height of Mount Everest above sea level. Now imagine a perfect globe of Mars, 30 cm across, the size of a big beach ball, with every mountain and crater shrunk by exactly the right amount.

How high would Olympus Mons stand on that globe? A few centimetres? A finger's width?

The answer is about 1 millimetre. You could barely feel it with your fingertip. A whole planet, shrunk to fit in your arms, is smoother than an orange. Working that out needs one idea: scaling, where every length is multiplied by the same number.

The idea

What a dilation is

A dilation (also called an enlargement, even when it shrinks) changes the size of a shape without changing its shape. It needs two things:

  • a centre of dilation, a fixed point \(O\);
  • a scale factor \(k\), a positive number.

Each point \(P\) moves along the ray from \(O\) through \(P\) to a new point \(P'\), so that \(P'\) is \(k\) times as far from \(O\) as \(P\) was:

\[OP' = k \times OP\]

With \(k = 2\) every point goes twice as far out; with \(k = \tfrac12\) every point comes halfway in.

OPQR P′Q′R′
Scale factor What happens
\(k > 1\) the image is bigger (an enlargement)
\(k = 1\) nothing moves
\(0 < k < 1\) the image is smaller (a reduction)

Dilations on the grid

When the centre is the origin, the rule is as simple as it could be: multiply both coordinates by \(k\).

\[(x, y) \to (kx,\ ky)\]

Why? The point \((x, y)\) is reached by going \(x\) across and \(y\) up. Going \(k\) times as far along the same ray means going \(k\) times as far across and \(k\) times as far up.

For a centre \(C\) that is not the origin, use the same trick as for rotations: measure from \(C\). If \(P\) is \(a\) across and \(b\) up from \(C\), then \(P'\) is \(ka\) across and \(kb\) up from \(C\).

What changes and what stays

Unlike the moves of the last lesson, a dilation is not an isometry: lengths change. But it changes them in the most orderly way possible.

  1. Every length is multiplied by \(k\). Not only distances from the centre: every side, every diagonal, every perimeter.
  2. Every angle stays the same. The shape is the same; only the size is different.
  3. Every line goes to a parallel line. Each side of the image is parallel to the matching side of the object.
  4. Every area is multiplied by \(k^2\). This one surprises people, so here is why.

Why lengths multiply by \(k\): take two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), with the centre at the origin. The step from \(P\) to \(Q\) is \(x_2 - x_1\) across and \(y_2 - y_1\) up. After the dilation, the step from \(P'\) to \(Q'\) is \(kx_2 - kx_1 = k(x_2 - x_1)\) across and \(k(y_2 - y_1)\) up. Every step is \(k\) times as long in each direction, so the straight-line distance is \(k\) times as long too, and its direction is unchanged (that is why lines stay parallel).

Why areas multiply by \(k^2\): cover the shape with tiny unit squares. A dilation turns each 1 by 1 square into a \(k\) by \(k\) square, whose area is \(k \times k = k^2\). Every little square gets \(k^2\) times bigger, so the whole area does. Double every length of a rectangle and you can fit four copies of the original inside the image, not two.

k = 1k = 2: 4×k = 3: 9×

(For solids, volumes multiply by \(k^3\), for the same reason with little cubes. Level 7 takes this further.)

Scale drawings, maps and blueprints

A scale drawing is a dilation of something real. Its scale is written as a ratio, drawing length : real length, using the same units on both sides. A scale of 1 : 50 means 1 cm on the drawing stands for 50 cm in real life, so the scale factor from the real thing to the drawing is \(k = \tfrac{1}{50}\).

  • House plans and blueprints often use 1 : 50 or 1 : 100.
  • Walking maps often use 1 : 25 000 or 1 : 50 000. At 1 : 25 000, 1 cm on the map is 25 000 cm, which is 250 m.
  • Model cars are often 1 : 18 or 1 : 43.

To go from the drawing to real life, multiply by the second number. To go from real life to the drawing, divide by it. Then convert units at the end: 100 cm = 1 m and 1000 m = 1 km, so 100 000 cm = 1 km.

Because areas scale by \(k^2\), a 1 : 50 plan shows areas \(50^2 = 2500\) times smaller than the real ones. Get this wrong and you buy a fiftieth of the carpet you need.

Worked examples

1. A dilation from the origin. Triangle \(A(1, 2)\), \(B(3, 2)\), \(C(3, 5)\), scale factor 3.

  1. Multiply every coordinate by 3: \(A'(3, 6)\), \(B'(9, 6)\), \(C'(9, 15)\).
  2. Check: \(AB\) runs from \(x = 1\) to \(x = 3\), length 2. \(A'B'\) runs from 3 to 9, length 6. That is \(3 \times 2\). \(BC\) is 3 long and \(B'C'\) is 9 long.
  3. The area was \(\tfrac12 \times 2 \times 3 = 3\); now it is \(\tfrac12 \times 6 \times 9 = 27 = 3^2 \times 3\).

2. A dilation from another centre. Centre \(C(1, 1)\), scale factor 2, point \(P(3, 4)\).

  1. From \(C\) to \(P\) is 2 across and 3 up.
  2. Double it: 4 across and 6 up.
  3. \(P' = (1 + 4,\ 1 + 6) = (5, 7)\).

3. A map. On a 1 : 25 000 map two lakes are 4.2 cm apart. The real distance is \(4.2 \times 25\,000 = 105\,000\) cm. Divide by 100 000 to get kilometres: 1.05 km.

4. A blueprint. On a 1 : 50 plan, a room measures 8 cm by 6 cm.

  1. Real size: \(8 \times 50 = 400\) cm \(= 4\) m, and \(6 \times 50 = 300\) cm \(= 3\) m.
  2. Real floor area: \(4 \times 3 = 12\) m². The drawing's area is \(8 \times 6 = 48\) cm².
  3. Check with \(k^2\): \(12\) m² is \(120\,000\) cm², and \(120\,000 \div 48 = 2500 = 50^2\).

5. Finding the centre and the scale factor. \(A(2, 1)\) goes to \(A'(5, 1)\) and \(B(2, 3)\) goes to \(B'(5, 9)\).

  1. \(AB\) is 2 long (from \(y = 1\) to \(y = 3\)). \(A'B'\) is 8 long. So \(k = 8 \div 2 = 4\).
  2. The centre lies on the line through \(A\) and \(A'\), which is the horizontal line \(y = 1\), and on the line through \(B\) and \(B'\). From \(B\) to \(B'\) is 3 across and 6 up, so that line climbs 2 for every 1 across. Going back from \(B(2, 3)\) by 1 across and 2 down reaches \((1, 1)\), which is on \(y = 1\). The centre is \((1, 1)\).
  3. Check: from \((1, 1)\) to \(A\) is 1 across; to \(A'\) is 4 across. Four times as far. Good.

6. The globe of Mars. Mars is about 6779 km across, and the globe is 30 cm across.

  1. Same units: 6779 km is \(677\,900\,000\) cm. The scale factor is \(k = 30 \div 677\,900\,000\), about 1 : 22.6 million.
  2. Olympus Mons is about 22 km \(= 2\,200\,000\) cm high. On the globe: \(2\,200\,000 \div 22\,600\,000 \approx 0.097\) cm, about 1 mm.
  3. The volcano is about 600 km wide, which on the globe is \(60\,000\,000 \div 22\,600\,000 \approx 2.7\) cm: a very low, very wide bump. Real Olympus Mons is like that too: its slopes are gentle, just very, very long.

Try it

  1. Dilate \((-2, 3)\) by scale factor 4 with centre at the origin.
  2. A 3 cm by 5 cm rectangle is enlarged by scale factor 2.5. Find the new side lengths and both areas. By what number was the area multiplied?
  3. On a 1 : 50 000 map, how far apart are two towns that are 7 km apart in real life?
  4. A model car is built at 1 : 18. The real car is 4.5 m long. How long is the model, in centimetres?
  5. A line segment 12 cm long becomes 3 cm long after a dilation. What is the scale factor?
  6. A photo 10 cm by 15 cm is enlarged so that its short side becomes 25 cm. How long is the long side?
  7. On a 1 : 200 plan, a square patio has an area of 6 cm². What is its real area in m²?
  8. Harder: a dilation has centre \((2, 0)\) and scale factor 3. Where does \((4, 1)\) go? Which point goes to \((5, 6)\)?
Answers
  1. \((4 \times (-2),\ 4 \times 3) = (-8, 12)\).
  2. \(3 \times 2.5 = 7.5\) cm and \(5 \times 2.5 = 12.5\) cm. Areas: \(3 \times 5 = 15\) cm² and \(7.5 \times 12.5 = 93.75\) cm². The area was multiplied by \(93.75 \div 15 = 6.25 = 2.5^2\).
  3. 7 km \(= 700\,000\) cm. On the map: \(700\,000 \div 50\,000 = 14\) cm.
  4. 4.5 m \(= 450\) cm. \(450 \div 18 = 25\) cm.
  5. \(k = 3 \div 12 = \tfrac14\). It is a reduction.
  6. \(k = 25 \div 10 = 2.5\), so the long side is \(15 \times 2.5 = 37.5\) cm.
  7. Areas scale by \(200^2 = 40\,000\). Real area \(6 \times 40\,000 = 240\,000\) cm². One m² is \(100 \times 100 = 10\,000\) cm², so this is \(240\,000 \div 10\,000 = 24\) m².
  8. From \((2, 0)\) to \((4, 1)\) is 2 across and 1 up; tripled, 6 across and 3 up: \((8, 3)\). For the second, go backwards: from \((2, 0)\) to \((5, 6)\) is 3 across and 6 up; divide by 3: 1 across and 2 up, so the point was \((3, 2)\).

Common mistake

Scaling areas by \(k\) instead of \(k^2\). Doubling the side of a square does not double its area; it makes it four times as big. A pizza 40 cm across has four times the area of one 20 cm across, not twice. Lengths scale by \(k\), areas by \(k^2\), volumes by \(k^3\).

Mixing units in a scale. "1 cm to 5 km" is fine as words, but as a ratio it must be in one unit: 1 : 500 000. Convert first, then multiply or divide.

Where it leads