Properties of Dilations
Aligned to G-SRT.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
A dilation with center O and scale factor k moves each point P to P' on ray OP so that OP' = k times OP. A dilation multiplies all distances by k but keeps every line parallel to its image, and it fixes the center. This is why dilated figures are similar, not congruent (unless k = 1).
Worked examples
Example 1
Dilate point A(2, 4) from the origin with scale factor 3. Multiply each coordinate by 3: A' = (3*2, 3*4) = (6, 12). The image lies on the ray from the origin through A, three times as far out. A segment from the origin to A is tripled in length, matching the scale factor.
Example 2
Dilate segment PQ with P(1, 2) and Q(4, 2) from center C(0, 0) by scale factor 1/2. Multiply each coordinate by 1/2: P'(0.5, 1) and Q'(2, 1). The original segment had length 3; the image has length 2 - 0.5 = 1.5, exactly half. Both the original and the image are horizontal, so the segment stayed parallel to itself.
Example 3
Common mistake: thinking a dilation with factor 1/2 shrinks toward the origin no matter what the center is. The center is the one fixed point, and it is not always the origin. Dilating A(6, 0) from center C(4, 0) by 1/2 moves A halfway to C: A' = (5, 0), not (3, 0). The rule is A' = C + k(A - C). Only when C is the origin does that collapse to simply multiplying the coordinates by k.
Practice questions
- Dilate B(1, -2) from the origin with scale factor 4. Give B'.
- A dilation with center O and scale factor 1/2 is applied to a segment of length 10. Find the image length.
- A line does not pass through the center of a dilation. What is true about the line and its image?
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Propiedades de las dilataciones
Una dilatación con centro O y factor de escala k mueve cada punto P a P' sobre el rayo OP de modo que OP' = k por OP. Una dilatación multiplica todas las distancias por k pero mantiene cada línea paralela a su imagen, y deja fijo el centro. Por eso las figuras dilatadas son semejantes, no congruentes (a menos que k = 1).
Ejemplo 1: Dilata el punto A(2, 4) desde el origen con factor de escala 3. Multiplica cada coordenada por 3: A' = (3*2, 3*4) = (6, 12). La imagen está sobre el rayo del origen que pasa por A, tres veces más lejos. Un segmento del origen a A se triplica, coincidiendo con el factor de escala.
Ejemplo 2: Dilata el segmento PQ con P(1, 2) y Q(4, 2) desde el centro C(0, 0) con factor de escala 1/2. Multiplica cada coordenada por 1/2: P'(0.5, 1) y Q'(2, 1). El segmento original medía 3; la imagen mide 2 - 0.5 = 1.5, exactamente la mitad. Tanto el original como la imagen son horizontales, así que el segmento quedó paralelo a sí mismo.
Ejemplo 3: Error común: creer que una dilatación de factor 1/2 encoge hacia el origen sin importar cuál sea el centro. El centro es el único punto fijo, y no siempre es el origen. Dilatar A(6, 0) desde el centro C(4, 0) con factor 1/2 mueve A hasta la mitad del camino hacia C: A' = (5, 0), no (3, 0). La regla es A' = C + k(A - C). Solo cuando C es el origen eso se reduce a multiplicar las coordenadas por k.
More Similarity, Right Triangles & Trigonometry lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required