Chapter 8-1 Pythagorean Theorem

Objectives  Students will be able to use the Pythagorean and its converse to find lengths in right triangles

Essential Understanding  If you know the lengths of any two sides of a right triangle, you can find the length of the third side by using the Pythagorean Theorem

Interesting Tidbits  What do you know about the Pythagorean Theorem?  More than 4000 years ago, the Babyloneans and the Chinese already knew that a triangle with the sides of 3, 4 and 5 must be a right triangle.  They used this knowledge to construct right angles.  They did this by dividing a string into twelve equal pieces and then laying it into a triangle so that one side is three, the second side four and the last side five sections long, they could easily construct a right angle.

Interesting Tidbits  This theorem is named after Pythagoras, a Greek mathematician who lived in the 500s B.C.  It is unclear if Pythagoras really was the first person to have found this relationship between the sides of right triangles, since no texts written by him were found. In fact, we can't even prove the guy lived. But the theorem a 2 + b 2 = c 2 got his name.  Another Greek, Euclid, wrote about the theorem about 200 years later in his book called "Elements".  In Euclid’s book, was the first known proof for the theorem. Now there are about 600 different proofs.

Pythagorean Theorem  In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

a 2 + b 2 = c 2

Pythagorean Triple  Nonzero, whole numbers, a, b, and c that satisfy the equation a 2 + b 2 = c 2  Common Pythagorean Triples:  3, 4, 5  5, 12, 13  How can you find another Pythagorean triple?

Find the Length of the Hypotenuse

Find the Length  The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse?  Is this a Pythagorean Triple?

Find the Length of a Leg

Find the Length  The hypotenuse of a right triangle has length 12. One leg has length 6. What is the length of the other leg? Write your answer in simplest radical form.

Find the Length  The size of a computer monitor is the length of its diagonal. You want to buy a 19 inch monitor that has a height of 11 inches. What is the width of the monitor? Round to the nearest tenth of an inch.

Is it a Right Triangle?  How do you think you determine if a triangle is a right triangle given the lengths of its sides?  A triangle has side lengths 16, 48, and 50. Is the triangle a right triangle?

Converse of the Pythagorean Theorem  Use to determine if a triangle is a right triangle  If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.

Is it a Right Triangle?  A triangle has side lengths 12.5, 30, and Is it a right triangle?

 How do you think we can determine if a triangle is obtuse or acute using the Pythagorean theorem?

Obtuse Triangle  If the square of the longest side of a triangle is greater than the sum of squares of the other sides of a triangle, the the triangle is obtuse.

Acute Triangle  If the square of the longest side of a triangle is less than the sum of squares of the other sides of a triangle, the the triangle is acute.

Classifying a Triangle  A triangle has side lengths 7, 8, and 9. Is it acute, obtuse, or right?  A triangle has side lengths 4, 9, and 12. Is it acute, obtuse, or right?

Homework  Pg. 495  #8 – 32 even, 38, 44  15 Problems