Photo apps, map zoom buttons, and movie projectors all use the same trick: they make a picture bigger or smaller without changing its shape. In geometry, that trick is called a dilation. In this chapter you will learn how dilations work, how to compute them on the coordinate plane, and how they lead to the big idea of similar figures. You will also review the angle facts about triangles and parallel lines that let you prove two triangles are similar.
1. What Is a Dilation?
A dilation is a transformation defined by a fixed point \(O\), called the center, and a positive number \(k\), called the scale factor. Each point \(P\) is sent to the point \(P'\) on the ray \(OP\) such that \[ OP' = k \cdot OP. \] The center \(O\) does not move.
Think of a flashlight shining on a wall: your hand is the original figure, the shadow is the image, and the flashlight is the center. The farther the wall is from the lamp, the bigger the shadow.
In the figure, the center is the origin \(O\) and the scale factor is \(2\). Each dashed ray starts at \(O\), goes through a vertex of triangle \(ABC\), and continues to the matching vertex of triangle \(A'B'C'\). Notice that \(OA' = 2 \cdot OA\), \(OB' = 2 \cdot OB\), and \(OC' = 2 \cdot OC\).
2. The Scale Factor
The scale factor \(k\) tells you what kind of dilation you have.
| Scale factor | Effect | Name |
|---|---|---|
| \(k > 1\) | The image is larger | Enlargement |
| \(0 < k < 1\) | The image is smaller | Reduction |
| \(k = 1\) | The image is identical | No change |
- Every length is multiplied by \(k\).
- Every angle keeps exactly the same measure.
- A segment is sent to a parallel segment (or to itself if its line passes through the center).
(a) A point \(P\) is \(6\) cm from the center \(O\). After a dilation with \(k = 2.5\), \(OP' = 2.5 \times 6 = 15\) cm.
(b) A side of a figure measures \(12\) inches. After a dilation with \(k = \dfrac{3}{4}\), the new side is \(\dfrac{3}{4} \times 12 = 9\) inches, so the figure shrinks.
A dilation multiplies lengths by \(k\); it does not add \(k\). Doubling a side of \(5\) cm gives \(10\) cm because \(5 \times 2 = 10\), not because you add \(2\). And a dilation never changes angle measures.
3. Dilations on the Coordinate Plane
When the center of the dilation is the origin \((0, 0)\), the rule is very simple.
The dilation with center \((0,0)\) and scale factor \(k\) sends \[ (x,\, y) \longrightarrow (kx,\, ky). \] If the center is a point \((a, b)\), use \((x, y) \to \big(a + k(x - a),\; b + k(y - b)\big)\).
- Write the coordinates of each vertex.
- Multiply both coordinates of each vertex by \(k\).
- Plot the new vertices and connect them in the same order.
- Check that each new side is \(k\) times the old one.
Dilate \(A(1, 1)\), \(B(3, 1)\), \(C(2, 3)\) by \(k = 2\) from the origin: \(A'(2, 2)\), \(B'(6, 2)\), \(C'(4, 6)\). Side \(AB = 2\) and \(A'B' = 4\), which is twice as long, as expected (see the first figure).
Dilate \(A(2, 2)\), \(B(8, 2)\), \(C(4, 6)\) by \(k = \dfrac{1}{2}\): \(A'(1, 1)\), \(B'(4, 1)\), \(C'(2, 3)\). The figure shows the smaller triangle sitting inside the larger one, with the same shape.
On my home planet we say: “The origin is the lamp, and the scale factor is the zoom.” Multiply both coordinates by the zoom and the whole picture follows!
4. Similar Figures
Two figures are similar if one can be obtained from the other by a sequence of rotations, reflections, translations, and dilations. We write \(\triangle ABC \sim \triangle DEF\).
If two figures are similar, then their corresponding angles are equal and their corresponding sides are proportional: all the ratios of matching sides are equal to the same number, the scale factor.
Triangle \(DEF\) has sides \(4, 6, 8\) and triangle \(ABC\) has sides \(6, 9, 12\). Compare the matching sides: \[ \dfrac{6}{4} = 1.5, \qquad \dfrac{9}{6} = 1.5, \qquad \dfrac{12}{8} = 1.5. \] All ratios are equal, so the triangles are similar with scale factor \(1.5\).
A rectangle \(4 \times 10\) and a rectangle \(6 \times 14\) are not similar, because \(\dfrac{6}{4} = 1.5\) but \(\dfrac{14}{10} = 1.4\).
5. Sequences of Transformations and Similarity
Rotations, reflections, and translations keep every length and angle: the image is congruent to the original. A dilation with \(k \neq 1\) changes the size. When a sequence of transformations includes at least one dilation, the image is similar but usually not congruent to the original.
Triangle \(ABC\) with \(A(1,1)\), \(B(4,1)\), \(C(1,3)\) is first reflected across the \(y\)-axis, which gives \(A_1(-1,1)\), \(B_1(-4,1)\), \(C_1(-1,3)\). Then it is dilated by \(k = 2\) from the origin, which gives \(A'(-2,2)\), \(B'(-8,2)\), \(C'(-2,6)\). Since the sequence has a reflection and a dilation, \(\triangle A'B'C' \sim \triangle ABC\) with scale factor \(2\): for instance \(AB = 3\) and \(A'B' = 6\).
6. Angles in Triangles and Parallel Lines
The three interior angles of any triangle add up to \(180^\circ\).
Why? Draw a line through one vertex parallel to the opposite side. The two angles formed next to the triangle are equal to two of the triangle’s angles (alternate interior angles), and together with the third angle they form a straight line, which measures \(180^\circ\).
When two parallel lines are cut by a transversal:
- corresponding angles are equal;
- alternate interior angles are equal;
- same-side interior angles add up to \(180^\circ\);
- vertical angles are always equal.
A triangle has angles \(47^\circ\), \(2x^\circ\), and \((x + 13)^\circ\). Then \(47 + 2x + x + 13 = 180\), so \(3x = 120\) and \(x = 40\). The angles are \(47^\circ\), \(80^\circ\), and \(53^\circ\), and indeed \(47 + 80 + 53 = 180\).
7. The Exterior Angle Theorem
If you extend one side of a triangle, the angle formed outside the triangle is an exterior angle.
An exterior angle of a triangle is equal to the sum of the two remote interior angles (the two interior angles that are not next to it).
Here is the proof. The exterior angle at \(C\) and the interior angle at \(C\) form a straight line, so the exterior angle is \(180^\circ - \angle C\). But the angle sum gives \(\angle A + \angle B = 180^\circ - \angle C\). Both quantities are equal! In the figure, \(48^\circ + 62^\circ = 110^\circ\), and also \(180^\circ - 70^\circ = 110^\circ\).
8. The Angle-Angle (AA) Criterion
If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
The reason is the angle sum: once two angles match, the third angles must match too (each is \(180^\circ\) minus the other two). All three pairs of angles are then equal, so one triangle is a scaled copy of the other. In the figure, both triangles have angles of \(50^\circ\), \(60^\circ\), and \(70^\circ\).
- Find two right triangles (or other triangles) that share two equal angles, for example because the sun’s rays are parallel.
- State that they are similar by AA.
- Write a proportion with matching sides.
- Solve for the unknown length.
A \(5\)-foot-tall person (about \(1.5\) m) casts an \(8\)-foot shadow. At the same time a tree casts a \(40\)-foot shadow. Both form right triangles and the sun’s angle is the same, so the triangles are similar by AA. Then \[ \dfrac{h}{40} = \dfrac{5}{8} \quad\Rightarrow\quad h = \dfrac{5 \times 40}{8} = 25. \] The tree is \(25\) feet tall, about \(7.6\) meters.
Key takeaways
- A dilation with center \(O\) and scale factor \(k\) sends \(P\) to \(P'\) on ray \(OP\) with \(OP' = k \cdot OP\).
- \(k > 1\) enlarges, \(0 < k < 1\) reduces; lengths are multiplied by \(k\) and angles do not change.
- On the coordinate plane with center at the origin: \((x, y) \to (kx, ky)\).
- Similar figures have equal corresponding angles and proportional corresponding sides.
- A sequence of rotations, reflections, translations, and dilations produces a similar figure.
- A triangle’s angles add up to \(180^\circ\); an exterior angle equals the sum of the two remote interior angles.
- Two triangles with two pairs of equal angles are similar (AA).
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