Pythagorean Triples

4 3 2 1 0In 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.

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

The Pythagorean Theorem,a2 + b2 = c2, 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 a2 + b2 = c2.

Pythagorean Triple: The set {3, 4, 5} is a Pythagorean

Triple. a2 + b2 = c2

32 + 42 = 52

9 + 16 = 25 25 = 25

Show that {5, 12, 13} is a Pythagorean triple. Always use the largest

value as c in the Pythagorean Theorem.

a2 + b2 = c2

52 + 122 = 132

25 + 144 = 169 169 = 169

Show that {2, 2, 5} is not a Pythagorean triple. a2 + b2 = c2

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}.

a2 + b2 = c2

62 + 82 = 102

36 + 64 = 100 100 = 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 Pythagoreantriple 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.