logging in or signing up The Pythagorean Theorem and its Converse aSGuest89101 Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 180 Category: Education License: All Rights Reserved Like it (0) Dislike it (0) Added: March 08, 2011 This Presentation is Public Favorites: 0 Presentation Description No description available. Comments Posting comment... Premium member Presentation Transcript Sec. 8-1 Pythagorean Theorem: Sec. 8-1 Pythagorean Theorem Objectives: 1) To use the Pythagorean Thm. 2) To use the converse of the Pythagorean Thm.Slide 2: Pythagoras (~580-500 B.C.) He was a Greek philosopher responsible for important developments in mathematics, astronomy and the theory of music. The Pythagorean Theorem is one of the most famous theorems in mathematics. The relationship it describes has been known for thousands of years.Slide 3: President Garfield may have been joking when he stated about his proof that, "we think it something on which the members of both houses can unite without distinction of the party." A nice feature of mathematical proofs is that they are not subject to political opinion. 20Slide 5: P ROVING THE P YTHAGOREAN T HEOREM THEOREM THEOREM 8-1 Pythagorean Theorem c 2 = a 2 + b 2 b a c 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 legs. (Converse is Theorem 8-2)Pythagorean Theorem: Pythagorean Theorem a 2 + b 2 = c 2 Legs Hyp Hypotenuse. – Longest side of a rt. Δ ** Only works for rt. Δ s c a b Legs * Sides that form the right Example 1: Find the missing side of the Δ.: Example 1: Find the missing side of the Δ . x 20 21 a 2 + b 2 = c 2 21 2 + 20 2 = x 2 441 + 400 = x 2 841 = x 2 √ 841 = √x 2 29 = xPythagorean Triple – Is a set of nonzero whole numbers that satisfy the equation: a2 + b2 = c2: Pythagorean Triple – Is a set of nonzero whole numbers that satisfy the equation: a 2 + b 2 = c 2 Examples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 ** Multiply each number in a pyth. Triple by the same whole number then the resulting numbers are pyth. Triples also. 6,8,10 15,20,25 …Math Review: Math Review Perfect Squares 1 2 = 1 2 2 = 4 3 2 = 9 4 2 = 16 5 2 = 25 6 2 = 36 7 2 = 49 8 2 = 64 9 2 = 81 Radicals √3 • √3 = √9 = 3 √40 = √4 • 10 = 2√10 √80 = √4•20 = 2√4•5 = 2•2√5 = 4√5Example 2: Solve for x and Simplify the radical: Example 2: Solve for x and Simplify the radical 20 x 8 a 2 + b 2 = c 2 x 2 + 8 2 = 20 2 x 2 + 64 = 400 x 2 = 336 √x 2 = √336 x = √(16)(21) x = 4√(21)Example 3: Find the area of ΔDCE: Example 3: Find the area of Δ DCE 12m 12m 20m A = ½ bh = ½ (20m)(6.6m) = 66m 2 a 2 + b 2 = c 2 10 2 + b 2 = 12 2 100 + b 2 = 144 b 2 = 44 b = √44 b = √(4)(11) b = 2√(11) D C E b = 6.6mExample 4: Are the following Δs, rt. Δ: Example 4: Are the following Δ s, rt. Δ 85 84 13 a 2 + b 2 = c 2 84 2 + 13 2 = 85 2 7056 + 169 = 7225 7225 = 7225 YESS ! 50 48 16 a 2 + b 2 = c 2 48 2 + 16 2 = 50 2 2304 + 256 = 2500 2560 ≠ 2500 Nope!Non-Right Δs: Non-Right Δ s Th(8-3) If the square of the length of the longest side of a Δ is greater than the sum of the squares of the lengths of the other 2 sides, the Δ is obtuse. If a 2 + b 2 < c 2 , then the Δ is obtuse.Slide 14: Th(8-4) If the square of the length of the longest side of a Δ is less than the sum of the squares of the lengths of the other 2 sides, then the Δ is acute. If a 2 + b 2 > c 2 , then the Δ is acute.Is the Δ Right, Obtuse, or Acute?: Is the Δ Right, Obtuse, or Acute? Ex. 5 Sides of 6, 11, 14 a 2 + b 2 = c 2 6 2 + 11 2 =14 2 36 + 121 = 196 157 < 196 Obtuse Ex. 6 Sides of 15, 13, 12 a 2 + b 2 = c 2 12 2 + 13 2 =15 2 144 + 169 = 225 313 > 225 AcuteEx. 7 Find AC and BC: Ex. 7 Find AC and BC The area of Δ ABC is 20ft 2 A C B h A = ½ bh 20ft 2 = ½ (10ft)h h = 4 To find AC a 2 + b 2 = c 2 2 2 + 4 2 = AC 2 4 + 16 = AC 2 AC = √ 20 AC = √ 4 •5 AC = 2 √ 5 To find BC a 2 + b 2 = c 2 4 2 + 8 2 =BC 2 16 + 64 = BC 2 80 = BC 2 BC = √ 16 • 5 BC = 4 √ 5 h 8ft 2ftSlide 17: Day 2 page 420, 18-29, 36,39,40,44 Day 1 page 420, 1-17 all You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.
The Pythagorean Theorem and its Converse aSGuest89101 Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 180 Category: Education License: All Rights Reserved Like it (0) Dislike it (0) Added: March 08, 2011 This Presentation is Public Favorites: 0 Presentation Description No description available. Comments Posting comment... Premium member Presentation Transcript Sec. 8-1 Pythagorean Theorem: Sec. 8-1 Pythagorean Theorem Objectives: 1) To use the Pythagorean Thm. 2) To use the converse of the Pythagorean Thm.Slide 2: Pythagoras (~580-500 B.C.) He was a Greek philosopher responsible for important developments in mathematics, astronomy and the theory of music. The Pythagorean Theorem is one of the most famous theorems in mathematics. The relationship it describes has been known for thousands of years.Slide 3: President Garfield may have been joking when he stated about his proof that, "we think it something on which the members of both houses can unite without distinction of the party." A nice feature of mathematical proofs is that they are not subject to political opinion. 20Slide 5: P ROVING THE P YTHAGOREAN T HEOREM THEOREM THEOREM 8-1 Pythagorean Theorem c 2 = a 2 + b 2 b a c 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 legs. (Converse is Theorem 8-2)Pythagorean Theorem: Pythagorean Theorem a 2 + b 2 = c 2 Legs Hyp Hypotenuse. – Longest side of a rt. Δ ** Only works for rt. Δ s c a b Legs * Sides that form the right Example 1: Find the missing side of the Δ.: Example 1: Find the missing side of the Δ . x 20 21 a 2 + b 2 = c 2 21 2 + 20 2 = x 2 441 + 400 = x 2 841 = x 2 √ 841 = √x 2 29 = xPythagorean Triple – Is a set of nonzero whole numbers that satisfy the equation: a2 + b2 = c2: Pythagorean Triple – Is a set of nonzero whole numbers that satisfy the equation: a 2 + b 2 = c 2 Examples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 ** Multiply each number in a pyth. Triple by the same whole number then the resulting numbers are pyth. Triples also. 6,8,10 15,20,25 …Math Review: Math Review Perfect Squares 1 2 = 1 2 2 = 4 3 2 = 9 4 2 = 16 5 2 = 25 6 2 = 36 7 2 = 49 8 2 = 64 9 2 = 81 Radicals √3 • √3 = √9 = 3 √40 = √4 • 10 = 2√10 √80 = √4•20 = 2√4•5 = 2•2√5 = 4√5Example 2: Solve for x and Simplify the radical: Example 2: Solve for x and Simplify the radical 20 x 8 a 2 + b 2 = c 2 x 2 + 8 2 = 20 2 x 2 + 64 = 400 x 2 = 336 √x 2 = √336 x = √(16)(21) x = 4√(21)Example 3: Find the area of ΔDCE: Example 3: Find the area of Δ DCE 12m 12m 20m A = ½ bh = ½ (20m)(6.6m) = 66m 2 a 2 + b 2 = c 2 10 2 + b 2 = 12 2 100 + b 2 = 144 b 2 = 44 b = √44 b = √(4)(11) b = 2√(11) D C E b = 6.6mExample 4: Are the following Δs, rt. Δ: Example 4: Are the following Δ s, rt. Δ 85 84 13 a 2 + b 2 = c 2 84 2 + 13 2 = 85 2 7056 + 169 = 7225 7225 = 7225 YESS ! 50 48 16 a 2 + b 2 = c 2 48 2 + 16 2 = 50 2 2304 + 256 = 2500 2560 ≠ 2500 Nope!Non-Right Δs: Non-Right Δ s Th(8-3) If the square of the length of the longest side of a Δ is greater than the sum of the squares of the lengths of the other 2 sides, the Δ is obtuse. If a 2 + b 2 < c 2 , then the Δ is obtuse.Slide 14: Th(8-4) If the square of the length of the longest side of a Δ is less than the sum of the squares of the lengths of the other 2 sides, then the Δ is acute. If a 2 + b 2 > c 2 , then the Δ is acute.Is the Δ Right, Obtuse, or Acute?: Is the Δ Right, Obtuse, or Acute? Ex. 5 Sides of 6, 11, 14 a 2 + b 2 = c 2 6 2 + 11 2 =14 2 36 + 121 = 196 157 < 196 Obtuse Ex. 6 Sides of 15, 13, 12 a 2 + b 2 = c 2 12 2 + 13 2 =15 2 144 + 169 = 225 313 > 225 AcuteEx. 7 Find AC and BC: Ex. 7 Find AC and BC The area of Δ ABC is 20ft 2 A C B h A = ½ bh 20ft 2 = ½ (10ft)h h = 4 To find AC a 2 + b 2 = c 2 2 2 + 4 2 = AC 2 4 + 16 = AC 2 AC = √ 20 AC = √ 4 •5 AC = 2 √ 5 To find BC a 2 + b 2 = c 2 4 2 + 8 2 =BC 2 16 + 64 = BC 2 80 = BC 2 BC = √ 16 • 5 BC = 4 √ 5 h 8ft 2ftSlide 17: Day 2 page 420, 18-29, 36,39,40,44 Day 1 page 420, 1-17 all