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Understanding the Pythagorean Theorem in Math

Learn about the Pythagorean Theorem, its proof, converse, and applications in calculating distances and working with special triangles.

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Understanding the Pythagorean Theorem in Math

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  1. Math 310 Section 11.3 The Pythagorean Theorem

  2. The Pythagorean Theorem In a right triangle the sum of the squares of the legs is equal to the square of the hypotenuse

  3. ? 3 4 Ex Given that the following triangle is a right triangle, and that the two given sides are the legs of the right triangle, what is the length of the third side.

  4. a b a b b a a b Proof of the Pythagorean Theorem Given that ABCD is a square and the congruent segments are indicated, prove the Pythagorean theorem.

  5. Converse of the Pythagorean Theorem If the sum of the squares of two sides of a triangle is equal to the square of the third side then the triangle is a right triangle and the right angel is opposite the largest side.

  6. 10 6 ? Ex What would the length of the third side of this triangle need to be for it to be a right triangle where the side of length 10 is the hypotenuse?

  7. Distance Formula The distance between two points, (x1, y1) and (x2, y2) is given by the following formula: D = ((x2 – x1)2 + (y2 – y1)2)1/2

  8. Ex Find the distance between the following sets of points: • (1, 1), (0, 0) • (2, 5), (-3, -1) • (1, -1), (4, -6)

  9. Origins of the Distance Formula Where does the distance formula come from? Well actually the distance formula come directly from the Pythagorean theorem. Can you see it?

  10. Special Triangles There are two triangles with their corresponding lengths and angles that should be memorized. • 30-60-90 Triangle • 45-45-90 Triangle

  11. 60° 2 1 90° 30° 31/2 30-60-90 Triangle

  12. 45° 21/2 1 90° 45° 1 45-45-90 Triangle

  13. Why Special Triangles Triangles with similar degree measurements seem to crop up quite often. Since two triangles with the same degree measurements are similar, using the similar triangles we can gain more information about a given triangle.

  14. 5(21/2) Ex What is the area of the following square?

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