1. Pythagorean Triples

2. 43210 In addition to level 3.0 and beyond what was taught in class, the student may:  Make connection with other concepts in math.  Make connection with other content areas.  Explain the relationship between the Pythagorean Theorem and the distance formula. The student will understand and apply the Pythagorean Theorem.  Prove the Pythagorean Theorem and its converse.  Apply the Pythagorean Theorem to real world and mathematical situations.  Find the distance between 2 points on a coordinate plane using the Pythagorean Theorem. The student will understand the relationship between the areas of the squares of the legs and area of the square of the hypotenuse of a right triangle.  Explain the Pythagorean Theorem and its converse.  Create a right triangle on a coordinate plane, given 2 points. With help from the teacher, the student has partial success with level 2 and level 3 elements.  Plot 3 ordered pairs to make a right triangle  Identify the legs and the hypotenuse of a right triangle Find the distance between 2 points on the coordinate grid (horizontal and vertical axis). Even with help, students have no success with the unit content. Focus 5 - Learning Goal #1: Students will understand and apply the Pythagorean Theorem.

3. Fact In a right triangle, the sides touching the right angle are called legs. The side opposite the right angle is the hypotenuse.

4. The Pythagorean Theorem, a 2 + b 2 = c 2, relates the sides of RIGHT triangles. a and b are the lengths of the legs and c is the length of the hypotenuse.

5. A Pythagorean Triple…  Is a set of three whole numbers that satisfy the Pythagorean Theorem. What numbers can you think of that would be a Pythagorean Triple? Remember, it has to satisfy the equation a 2 + b 2 = c 2.

6. Pythagorean Triple:  The set {3, 4, 5} is a Pythagorean Triple.  a 2 + b 2 = c 2  3 2 + 4 2 = 5 2  9 + 16 = 25  25 = 25

7. Show that {5, 12, 13} is a Pythagorean triple.  Always use the largest value as c in the Pythagorean Theorem.  a 2 + b 2 = c 2  5 2 + 12 2 = 13 2  25 + 144 = 169  169 = 169

8. Show that {2, 2, 5} is not a Pythagorean triple.  a 2 + b 2 = c 2  2 2 + 2 2 = 5 2  4 + 4 = 25  8 ≠ 25  Showing that three numbers are a Pythagorean triple proves that the triangle with these side lengths will be a right triangle.

9. How to find more Pythagorean Triples  If we multiply each element of the Pythagorean triple, such as {3, 4, 5} by another integer, like 2, the result is another Pythagorean triple {6, 8, 10}.  a 2 + b 2 = c 2  6 2 + 8 2 = 10 2  36 + 64 = 100  100 = 100

10. By knowing Pythagorean triples, you can quickly solve for a missing side of certain right triangles.  Find the length of side b in the right triangle below.  Use the Pythagorean triple {5, 12, 13}.  The length of side b is 5 units.

11. Find the length of side a in the right triangle below.  It is not obvious which Pythagorean triple these sides represent.  Begin by dividing the given sides by their GCF (greatest common factor)  { ___, 16, 20} ÷ 4  { ___, 4, 5}  We see this is a {3, 4, 5} Pythagorean triple.  Since we divided by 4, we must now do the opposite and multiple by 4.  {3, 4, 5} 4 = {12, 16, 20}  Side a is 12 units long.