Right Triangles A triangle is the simplest polygon in a plane, consisting of three line segments There are many uses of the triangle, especially in construction work such as bridge building, radio/cell phone towers, and airplane wings. A RIGHT triangle is a triangle that has one right (90°) angle.

Label sides of right triangle In right triangle ABC below, the two sides of the triangle that form the right angle, A and C, are called the legs of the right triangle. B, the side opposite the right angle, is called the hypotenuse. The hypotenuse is the longest side of a right triangle.

LABEL Name the two legs of the right triangle below. Side ___________ and ______________ Name the hypotenuse of the right triangle below. ________________ D F E

Pythagorean Theorem More than 2000 years ago, the Greek mathematician Pythagoras demonstrated the following property of the right triangle: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Leg² + Leg² = Hypotenuse² OR a² + b² = c² where c is the hypotenuse

Pythagorean Triples To find the length of the hypotenuse of a right triangle whose legs measure 3 and 4 is as follows: 3² + 4² = Hypotenuse² = Hypotenuse² 25= Hypotenuse² √25 =√Hypotenuse² 5=Hypotenuse

Pythagorean Triples When three integers are so related that the sum of the squares of two of them is equal to the square of the third, the set of three integers is called a Pythagorean triple. Our example, (3,4,5) is a Pythagorean Triple because the numbers satisfy the Pythagorean Theorem. In general, if we multiply each number of a Pythagorean triple by the same positive number, then the new triple created is also a Pythagorean triple. In other words, the three sides are in the same ratio. For example, if we multiply each side of a 3,4,5 right triangle by 2, the new sides of a right triangle would be 6,8,10. If we multiply each side by 10, the new sides of of the right triangle would be 30,40,50. Other examples of Pythagorean triples that occur frequently are (5,12,13) and (8,15,17)

Pythagorean Theorem Applications 1.Using the diagram of the right triangle, the hypotenuse has a length of 20 and the length of side a is 16. Find the length of the other leg. 2. Using the diagram of the right triangle, side a is 9 and side b is 40. Find the length of the hypotenuse. 3. Using the diagram of the right triangle, side b is 6 and the hypotenuse is 10. Find the length of side a.

More Practice Using the Pythagorean Theorem One end of a 10 foot ladder is 4 feet from the base of a wall. How high on the wall does the top of the ladder touch?

Prove