Side-Splitter and Pythagorean Proofs
Aligned to G-SRT.B.4 — Common Core State Standards for Mathematics.
What this lesson teaches
Similarity proves major theorems. The side-splitter theorem: a line parallel to one side of a triangle divides the other two sides proportionally. The Pythagorean theorem can be proved by dropping an altitude to the hypotenuse, which creates two smaller triangles similar to the whole.
Worked examples
Example 1
Prove a^2 + b^2 = c^2 using similarity. In right triangle with legs a, b and hypotenuse c, drop an altitude from the right angle to the hypotenuse, splitting it into segments p and q (p + q = c). Each small triangle is similar to the original, giving a^2 = c*p and b^2 = c*q. Add: a^2 + b^2 = c*p + c*q = c(p + q) = c*c = c^2.
Example 2
Apply the side-splitter theorem. In triangle ABC, a line parallel to BC cuts AB at D and AC at E. If AD = 4, DB = 6 and AE = 6, find EC. The parallel line creates a similar triangle, so AD/DB = AE/EC. Substitute: 4/6 = 6/EC. Cross-multiply: 4 × EC = 36, so EC = 9.
Example 3
Common mistake: splitting the wrong pair of lengths. The theorem relates the two PIECES of each side, or the two WHOLE sides — but you must stay consistent. AD/DB = AE/EC is correct (piece to piece), and AD/AB = AE/AC is correct (piece to whole), but AD/DB = AE/AC mixes the two and is false. Here that mix would give 4/6 = 6/AC, or AC = 9, when AC is really 6 + 9 = 15.
Practice questions
- A line parallel to the base splits the left side of a triangle into 4 and 6. If it splits the right side's top part as 6, find the bottom part.
- In a right triangle the altitude to the hypotenuse creates segments 4 and 9. Find the altitude's length (geometric mean).
- A right triangle has legs 9 and 12. Use a^2 + b^2 = c^2 to find the hypotenuse.
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
Demostraciones del divisor de lados y Pitágoras
La semejanza demuestra teoremas importantes. El teorema del divisor de lados: una línea paralela a un lado de un triángulo divide los otros dos lados proporcionalmente. El teorema de Pitágoras se demuestra trazando una altura a la hipotenusa, que crea dos triángulos menores semejantes al total.
Ejemplo 1: Demuestra a^2 + b^2 = c^2 usando semejanza. En un triángulo rectángulo con catetos a, b e hipotenusa c, traza una altura desde el ángulo recto a la hipotenusa, partiéndola en segmentos p y q (p + q = c). Cada triángulo menor es semejante al original, dando a^2 = c*p y b^2 = c*q. Suma: a^2 + b^2 = c*p + c*q = c(p + q) = c*c = c^2.
Ejemplo 2: Aplica el teorema de la recta paralela (Tales). En el triángulo ABC, una recta paralela a BC corta a AB en D y a AC en E. Si AD = 4, DB = 6 y AE = 6, halla EC. La recta paralela crea un triángulo semejante, así que AD/DB = AE/EC. Sustituye: 4/6 = 6/EC. Multiplica en cruz: 4 × EC = 36, entonces EC = 9.
Ejemplo 3: Error común: dividir el par de longitudes equivocado. El teorema relaciona las dos PARTES de cada lado, o los dos lados COMPLETOS — pero hay que ser consistente. AD/DB = AE/EC es correcto (parte con parte) y AD/AB = AE/AC es correcto (parte con todo), pero AD/DB = AE/AC mezcla ambos y es falso. Aquí esa mezcla daría 4/6 = 6/AC, o sea AC = 9, cuando AC en realidad es 6 + 9 = 15.
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