Pythagorean Triples 4 Focus 5 Learning Goal 1

Pythagorean Triples

4 Focus 5 - Learning Goal #1: Students will understand apply the Pythagorean Theorem. 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. 3 2 1 0 The student will understand 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.

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

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.

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.

Pythagorean Triple: The set {3, 4, 5} is a Pythagorean Triple. a 2 + b 2 = c 2 32 + 4 2 = 5 2 9 + 16 = 25

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 52 + 122 = 132 25 + 144 = 169

Show that {2, 2, 5} is not a Pythagorean triple. a 2 + b 2 = c 2 22 + 22 = 52 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.

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 62 + 82 = 102 36 + 64 = 100

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.

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.