Similarity Transformations
Aligned to G-SRT.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
Two figures are similar when a sequence of rigid motions and a dilation maps one onto the other. For triangles this means corresponding angles are equal and corresponding sides are proportional. The common ratio of the sides is the scale factor.
Worked examples
Example 1
Triangle ABC has sides 3, 4, 5. Triangle DEF has sides 6, 8, 10 with equal corresponding angles. Check proportionality: 6/3 = 2, 8/4 = 2, 10/5 = 2. All ratios equal 2, so a dilation of factor 2 (with rigid motions) maps ABC onto DEF. The triangles are similar with scale factor 2.
Example 2
A photo is 4 inches by 6 inches and is enlarged to 10 inches wide. Is the enlargement similar to the original? Check the scale factor on the known side: 10/4 = 2.5. For similarity the other side must be 6 × 2.5 = 15 inches. If the frame is 10 by 14, the ratios are 2.5 and 14/6 = 2.33 — not equal, so the picture is NOT similar to the original and will be stretched.
Example 3
Common mistake: adding the same amount to every side instead of multiplying. Growing a 3-4-5 triangle by adding 2 to each side gives 5-6-7. The ratios are 5/3 = 1.67, 6/4 = 1.5 and 7/5 = 1.4 — all different, so it is not similar, and the right angle is gone. Similarity comes from a dilation, and a dilation MULTIPLIES. Equal differences preserve nothing; equal ratios preserve shape.
Practice questions
- Triangle 1 has sides 5, 7, 9. Triangle 2 is similar with scale factor 3. List its side lengths.
- Two similar triangles have corresponding sides 12 and 4. What is the scale factor from the small to the large?
- If two triangles are similar, what must be true about each pair of corresponding angles?
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
Transformaciones de semejanza
Dos figuras son semejantes cuando una secuencia de movimientos rígidos y una dilatación mapea una sobre la otra. Para triángulos esto significa que los ángulos correspondientes son iguales y los lados correspondientes son proporcionales. La razón común de los lados es el factor de escala.
Ejemplo 1: El triángulo ABC tiene lados 3, 4, 5. El triángulo DEF tiene lados 6, 8, 10 con ángulos correspondientes iguales. Verifica la proporcionalidad: 6/3 = 2, 8/4 = 2, 10/5 = 2. Todas las razones son 2, así que una dilatación de factor 2 (con movimientos rígidos) mapea ABC sobre DEF. Los triángulos son semejantes con factor de escala 2.
Ejemplo 2: Una foto mide 4 pulgadas por 6 pulgadas y se amplía a 10 pulgadas de ancho. ¿Es la ampliación semejante a la original? Comprueba el factor de escala en el lado conocido: 10/4 = 2.5. Para que haya semejanza, el otro lado debe medir 6 × 2.5 = 15 pulgadas. Si el marco es de 10 por 14, las razones son 2.5 y 14/6 = 2.33 — distintas, así que la imagen NO es semejante a la original y quedará deformada.
Ejemplo 3: Error común: sumar la misma cantidad a cada lado en vez de multiplicar. Agrandar un triángulo 3-4-5 sumando 2 a cada lado da 5-6-7. Las razones son 5/3 = 1.67, 6/4 = 1.5 y 7/5 = 1.4 — todas distintas, así que no es semejante, y el ángulo recto desapareció. La semejanza viene de una dilatación, y una dilatación MULTIPLICA. Las diferencias iguales no conservan nada; las razones iguales conservan la forma.
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