The Pythagorean Theorem and Its Converse

History of the Theorem Pythagoras of Samos was a Greek philosopher responsible for many important developments in mathematics! But, rumour has it Pythagoras’ Theorem was known to the Babylonians some 1000 years before Pythagoras. However, we all believe he was the first person to prove the theorem and that is why the theorem takes his name.

Parts of a Right Triangle In a right triangle, the side opposite the right angle is called the hypotenuse. It is the longest side. The other two sides are called the legs.

The Pythagorean Theorem Pythagorean Theorem: If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b2 = c2

Pythagorean Triples A Pythagorean triple is a set of nonzero whole numbers that satisfy the Pythagorean Theorem. Some common Pythagorean triples include: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 If you multiply each number in the triple by the same whole number, the result is another Pythagorean triple!

Pythagoras Questions 1 2 x 3 cm 4 cm Pythagorean triple x 5 cm 12 cm         Pythagorean triple     5 cm 12 cm x 2       Pythagorean triple

x ≈ 6.32 cm x ≈ 21.11 cm Pythagoras Questions: Finding a leg measure 3   x m 9 m 11m 3         x ≈ 6.32 cm Another method for finding a leg measure     11 cm x cm 23.8 cm 4   x ≈ 21.11 cm

Applications of Pythagoras Find the diagonal of the rectangle 6 cm 9.3 cm 1 d         d = 11.07 cm

x ≈ 6.51 cm Perimeter = 2(6.51+4.3) ≈ 21.62 cm 2 A rectangle has a width of 4.3 cm and a diagonal of 7.8 cm. Find its perimeter. 2 7.8 cm 4.3 cm x cm           x ≈ 6.51 cm Perimeter = 2(6.51+4.3) ≈ 21.62 cm therefore

Finding the Length of the Hypotenuse What is the length of the hypotenuse of ABC? Do the sides form a Pythagorean triple?

 The legs of a right triangle have lengths 10 and 24  The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse? Do the sides form a Pythagorean triple?

Finding the Length of a Leg What is the value of x? Express your answer in simplest radical form.

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

Triangle Classifications Converse of the Pythagorean Theorem: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If c2 = a2 + b2, than ABC is a right triangle.

The converse of the Pythagorean Theorem can be used to categorize triangles. If c2 = a2 + b2, then triangle ABC is a right triangle. If c2 > a2 + b2, then triangle ABC is an obtuse triangle. If c2 < a2 + b2, then triangle ABC is an acute triangle.

Triangle Inequality The triangle is obtuse 38, 77, 86 c2 ? a2 + b2 862 ? 382 + 772 7396 ? 1444 + 5929 7396 > 7373 The triangle is obtuse

Triangle Inequality The triangle is acute 10.5, 36.5, 37.5 c2 ? a2 + b2 37.52 ? 10.52 + 36.52 1406.25 ? 110.25 + 1332.25 1406.25 < 1442.5 The triangle is acute

4,7,9 9²__4² + 7² 81__16 + 49 81 > 65 OBTUSE greater

5,5,7 7² __5² + 5² __ 25 +25 49 < 50 ACUTE Less than