The Pythagorean Theorem
Aligned to 8.G.B.6 — Common Core State Standards for Mathematics.
What this lesson teaches
In any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a^2 + b^2 = c^2. The converse is also true: if a^2 + b^2 = c^2 for a triangle's sides, then it is a right triangle.
Worked examples
Example 1
A right triangle has legs 3 and 4, so c^2 = 3^2 + 4^2 = 9 + 16 = 25 and c = √25 = 5. To test the converse: since 3^2 + 4^2 = 5^2, a triangle with sides 3, 4, 5 must be a right triangle.
Example 2
Legs 5 and 12: c² = 25 + 144 = 169, so c = 13. The 5-12-13 triple is worth remembering alongside 3-4-5.
Example 3
Test 6, 7, 10 with the converse: 36 + 49 = 85, but 10² = 100. Since 85 ≠ 100, that triangle is not right.
Practice questions
- Find the hypotenuse of a right triangle with legs 6 and 8.
- A right triangle has a leg of 5 and a hypotenuse of 13. Find the other leg.
- Is a triangle with sides 8, 15, 17 a right triangle? Justify using the converse.
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
El Teorema de Pitágoras
En todo triángulo rectángulo, el cuadrado de la hipotenusa es igual a la suma de los cuadrados de los dos catetos: a^2 + b^2 = c^2. El recíproco también es cierto: si a^2 + b^2 = c^2 para los lados de un triángulo, entonces es un triángulo rectángulo.
Ejemplo 1: Un triángulo rectángulo tiene catetos 3 y 4, así que c^2 = 3^2 + 4^2 = 9 + 16 = 25 y c = √25 = 5. Para probar el recíproco: como 3^2 + 4^2 = 5^2, un triángulo con lados 3, 4, 5 debe ser rectángulo.
Ejemplo 2: Catetos 5 y 12: c² = 25 + 144 = 169, así que c = 13. Vale la pena recordar la terna 5-12-13 junto con la 3-4-5.
Ejemplo 3: Prueba 6, 7, 10 con el recíproco: 36 + 49 = 85, pero 10² = 100. Como 85 ≠ 100, ese triángulo no es rectángulo.
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