Algebra Honors (Periods 6 & 7)

The Pythagorean Theorem 11-6

The Pythagorean Theorem can be used to find the lengths of a right triangle. The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side. The other two sides of a right triangle are called the legs of the triangle.

In any right triangle the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs.  a² + b² = c²

Example:

The length of one side of a right triangle I 28 cm the length of the hypotenuse is 53 cm. Write and solve an equation for the length of the unknown side.

a² + b² = c²       a² = c² –b²            

Pythagorean Triples

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

Constructions:

To draw a line segment with a length of √2 , draw a right triangle with legs of length 1 unit . Using that length You can construct a segment  √3 units long… and so on

Converse of the Pythagorean Theorem

If the sum of the squares of the lengths of the two shorter sides of a triangle is equal to the square of the length of the longest side of the triangle, then the triangle is a right triangle.

The Distance Formula

The distance between two pints on the x-axis 
(or a line parallel to that axis)  is the absolute value of the
difference between their x-coordinates.

The distance between two points on the y-axis
(or a line parallel to that axis) is the absolute value of the
difference between their y-coordinates.

To find the distance between two points NOT on an axis or a line parallel to either axis, use the Pythagorean Theorem.

This can be generalized as the Distance Formula

For any points P₁(x₁, x₂) and P₂ (x₂, y₂)