Lesson 8-3 The Converse of the Pythagorean Theorem (page 295) Essential Question How can you determine whether a triangle is acute, right, or obtuse?

The converse of a theorem is not necessarily true, but the converse of the Pythagorean Theorem is true!

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. Theorem 8-3 Given: ∆ ABC with c 2 = a 2 + b 2 Prove: ∆ ABC is a right triangle. A C B ab c

Example #1 (a) Are the lengths 4, 7, and 9 sides of a right triangle? _________ NO ∴ not a right triangle!

Example #1 (b) Are the lengths 20, 21, and 29 sides of a right triangle? _________ YES ∴ this is a right triangle!

Example #1 (c) Are the lengths 0.8, 1.5, and 1.7 sides of a right triangle? _________ YES ∴ this is a right triangle!

A triangle with sides of 3, 4, and 5 is a right triangle because … This is a very common triangle, called a ___________ triangle. Any triangle with sides 3n, 4n, and 5n, where n > 0, is also a right triangle, because …. Multiples of any 3 lengths that form a right triangle will also form __________ triangles. These groups of 3 lengths are called _____________________. 5 2 = ⇒ 25 = (5n) 2 = (4n) 2 + (3n) 2 25n 2 = 16n 2 + 9n 2 25n 2 = 25n 2 right Pythagorean Triples

This is a triangle.

multiply each by 2 ⇒ 2·3, 2·4, 2·5 ⇒ 6, 8, 10 Some Common Right Triangle Lengths AKA: Pythagorean Triples 3, 4, 5 multiply each by 3 ⇒ 3·3, 3·4, 3·5 ⇒ 9, 12, 15 multiply each by 0.1 ⇒.1·3,.1·4,.1·5 ⇒ 0.3, 0.4, 0.5 multiply each by 10 ⇒ 10·3, 10·4, 10·5 ⇒ 30, 40, 50

multiply each by 2 ⇒ 2·5, 2·12, 2·13 ⇒ 10, 24, 26 Some Common Right Triangle Lengths AKA: Pythagorean Triples 5, 12, 13 multiply each by 3 ⇒ 3·5, 3·12, 3·13 ⇒ 15, 36, 39 multiply each by 0.1 ⇒.1·5,.1·12,.1·13 ⇒ 0.5, 1.2, 1.3 multiply each by 10 ⇒ 10·5, 10·12, 10·13 ⇒ 50, 120, 130

multiply each by 2 ⇒ 2·8, 2·15, 2·17 ⇒ 16, 30, 34 Some Common Right Triangle Lengths AKA: Pythagorean Triples 8, 15, 17 multiply each by 3 ⇒ 3·8, 3·15, 3·17 ⇒ 24, 45, 51 multiply each by 0.1 ⇒.1·8,.1·15,.1·17 ⇒ 0.8, 1.5, 1.7 multiply each by 10 ⇒ 10·8, 10·15, 10·17 ⇒ 80, 150, 170

multiply each by 2 ⇒ 2·7, 2·24, 2·25 ⇒ 14, 48, 50 Some Common Right Triangle Lengths AKA: Pythagorean Triples 7, 24, 25 multiply each by 3 ⇒ 3·7, 3·24, 3·25 ⇒ 21,72, 75 multiply each by 0.1 ⇒.1·7,.1·24,.1·25 ⇒ 0.7, 2.4, 2.5 multiply each by 10 ⇒ 10·7, 10·24, 10·25 ⇒ 70, 240, 250

Is 11, 60, 61 a Pythagorean Triple? ∴ 11, 60, 61 is a Pythagorean Triple!

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. Theorem 8-4 If c 2 < a 2 + b 2, then m ∠ C < 90º and ∆ABC is acute. A C B ab c

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. Theorem 8-5 If c 2 > a 2 + b 2, then m ∠ C > 90º and ∆ABC is obtuse. A C B ab c

Example #2 (1) If a triangle is formed by the lengths 8, 12, and 13, is it acute, right, or obtuse ? ∴ an acute triangle! Longest side Longest side

Example #2 (2) If a triangle is formed by the lengths 4, 4, and 7, is it acute, right, or obtuse ? ∴ an obtuse triangle! Longest side Longest side

Example #2 (3) If a triangle is formed by the lengths 8, 9, and 12, is it acute, right, or obtuse ? ∴ an acute triangle! Longest side Longest side

Example #2 (4) If a triangle is formed by the lengths 8, 11, and 15, is it acute, right, or obtuse ? ∴ an obtuse triangle! Longest side Longest side

Example #2 (5) If a triangle is formed by the lengths 4, 5, and 6, is it acute, right, or obtuse ? ∴ an acute triangle! Longest side Longest side

Example #2 (6) If a triangle is formed by the lengths 8, 9, and 17, is it acute, right, or obtuse ? An obtuse triangle? HOW? Longest side Longest side

Example #2 (6) If a triangle is formed by the lengths 8, 9, and 17, is it acute, right, or obtuse ? An obtuse triangle? HOW? NO! Not even a ∆. Remember?

If c 2 > a 2 + b 2, then ∆ is an OBTUSE TRIANGLE. If c 2 < a 2 + b 2, then ∆ is an ACUTE TRIANGLE. If c 2 = a 2 + b 2, then ∆ is a RIGHT TRIANGLE. Theorem Summary How can you determine whether a triangle is acute, right, or obtuse?

Assignment Written Exercises on page 297 DO NOW: 1 to 9 all numbers GRADED: 11 to 14 all numbers ~ #16 BONUS! ~ How can you determine whether a triangle is acute, right, or obtuse?