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 example
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.
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: 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.
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