Geometry 8 3 Converse of the Pythagorean Theorem

Geometry 8. 3 Converse of the Pythagorean Theorem

Theorem: Converse of the Pythagorean Theorem If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. If c² = a² + b² Rt. ∆ c b a

Directions: If a triangle is formed with sides having the lengths given, it is a right triangle? 1. 4, 7, 9 2. 20, 21, 29 3. 4. 0. 8, 1. 5, 1. 7

• Because 3² + 4²= 5², a ∆ with sides 3, 4, 5 and multiples of 3, 4, 5 is always a RIGHT TRIANGLE. • Multiples of any three lengths that form a Rt. ∆ will also form Rt. ∆’s. • The special groups of 3 lengths are called: Pythagorean Triples

Pythagorean Triples 3, 4, 5 6, 8, 10 9, 12, 15 12, 16, 20 15, 20, 25 etc. 5, 12, 13 10, 24, 15, 36, 20, 48, 25, 60, etc. 26 39 52 65 8, 15, 17 16, 30, 34 24, 45, 51 32, 60, 68 etc. 7, 24, 25 14, 48, 50 21, 72, 75 28, 96, 100 etc. Memorize the 4 special triples at the top. Use them to save time and effort.

Using Pythagorean Triples 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 Quickly find the value of x. 1. 48 14 2. x 7, 24, 25 14, 48, x x = 50 x 34 8, 15, 17 16, x, 34 x = 30 16

Theorem If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is an obtuse triangle. If c² > a² + b² Obtuse ∆ b ob c tu se a

Theorem If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is an acute triangle. If c² < a² + b² Acute ∆ c b ac u te a

Directions: If a triangle is formed with the given lengths, is it acute, right, or obtuse? 5. 8, 9, 12 6. 9. 3, 10. 8, 11, 15 7. 8, 13, 20 11. 4, 5, 6 8. 5, 7, 12. 5, 5,

Homework pg. 297 #1 -18 pg. 293 #23 -31 Odd

Exercises If a ∆ is formed with sides having the lengths given, is it a right ∆ ? 1. 4, 7, 9 Answers: 3. √ 2, 2, √ 5 2. Yes 4. Yes

Exercises If a ∆ is formed with the given lengths, is it acute, right, or obtuse? 7. 8, 13, 20

Exercises If a ∆ is formed with the given lengths, is it acute, right, or obtuse? 5. 8, 9, 12 6. √ 5, √ 10

Answers to Notes Packet Exercises 8 - 12 8. 9. 10. 11. 12. right obtuse acute obtuse 74 = 25 + 49 16 = 4 + 12 225 > 64 + 121 36 < 16 + 25 75 > 25 + 25

Review: Theorem If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. ~ a b a ~ b

Review: Corollary 1 piece of hypotenuse altitude X altitude = other piece of hypotenuse Y A Z

Review: Corollary 2 hypotenuse leg piece of hyp. adj. to leg = X Y A Z

Review: Corollary 2 hypotenuse leg piece of hyp. adj. to leg = X Y A Z