# 3.1 – The Pythagorean Theorem

## Objectives

• Derive the Pythagorean theorem $a^2+b^2=c^2$ from the relationship between right triangles and squares.
• Use the Pythagorean theorem to find the values of missing sides in a right triangle.
• Prove that triangles are right triangles using the converse of the Pythagorean theorem.
• Define and give examples of Pythagorean triples.
• Use the Pythagorean theorem to solve real-world problems.

## Key Terms

• Hypotenuse – The side opposite from the right angle in a right triangle.
• It is the triangle’s longest side.
• Leg – Either of the two shorter sides of a right triangle.
• Always touches the right angle.
• Pythagorean Theorem – The theorem that relates the side lengths of a right triangle.
• The theorem states that the square of the hypotenuse equals the sum of the squares of the legs: $a^2+b^2=c^2$
• Pythagorean Triple – A set of three whole numbers, a, b, and c, that satisfies the equation $a^2+b^2=c^2$.
• If the side lengths of a triangle form a Pythagorean triple, it is a right triangle.

## Notes

Right Triangles
• Right Triangles are:
• Triangles with one right angle (an angle measuring 90°).
• The other two angles in a right triangle are always acute (less than 90°).

• Remember: the area of a square is $(side)^2$

Pythagorean Theorem
• Pythagorean Theorem
• Pythagorean theorem: If a triangle is a right triangle, then the square of its longest side equals the sum of the squares of its other two sides.
• Converse: If the square of the longest side of a triangle equals the sum of the squares of its other two sides, then the triangle is a right triangle
• Solve for a missing side of a right triangle
• 2. Plug in the side lengths you know.
• 3. Solve the equation for the unknown side length

• Examples

• Tests for Right Triangles

• Common Pythagorean Triples
• How to Find a Pythagorean Triple: