Geometry: Understanding the Pythagorean Theorem and Triple Sets

Explore the fundamentals of Geometry with a focus on the Pythagorean Theorem and Pythagorean Triple Sets. Learn how to classify triangles, solve right triangle components, and identify Pythagorean Triple integer sets. Dive into practical examples and applications of the theorem's converse, followed by in-depth explanations and demonstrations. Enhance your knowledge and problem-solving skills in Geometry through this comprehensive study material.

• Geometry
• Pythagorean Theorem
• Triangles
• Pythagorean Triple
• Mathematics

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1. Before you start, go to Slide Show and click Play from start . Hit enter to go to the next slide. Thank you.

2. Warm-up: Do this on a separate piece of paper. Take the paper home with you to study. Quiz Target 8D next class. RQ ST 1.5 ST=6 \ST =3 =QP TP 12 XW ZW 15 21= 360-15XY = 21XY 360 =36XY \XY =10 =XY ZY AC BD=CE AC 21=16 \AC = 28 XY DF 24- XY 12

3. Geometry Geometry POINTS, LINES AND PLANES 9 9- -1 The Pythagorean Theorem 1 The Pythagorean Theorem Page 244 Page 244 Target 9A. I CAN use the converse of the Pythagorean Theorem to classify a right, acute or obtuse triangle. Target 9B. I CAN solve and identify the parts of a right triangle.

4. Geometry Geometry POINTS, LINES AND PLANES 9 9- -1 The Pythagorean Theorem 1 The Pythagorean Theorem Page 244 Page 244 Theorem 9-1 The Pythagorean Theorem. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length of the legs. Notes Ex: Find the value of x. AC2= AB2+BC2 1252=352+ x2 15625=1225+ x2 14400 = x2 14400 = x \120 = x

5. Geometry Geometry Page 244 Page 244 POINTS, LINES AND PLANES 9 9- -1 The Pythagorean Theorem 1 The Pythagorean Theorem Pythagorean Triple. A Pythagorean Triple is a set of three positive integers (whole numbers) a, b and c which satisfy the Pythagorean Theorem.

6. Geometry Geometry Page 244 Page 244 POINTS, LINES AND PLANES 9 9- -1 The Pythagorean Theorem 1 The Pythagorean Theorem Notes Ex: Find the value of x. Then tell whether the side lengths form a Pythagorean Triple. a) BC2= AB2+ AC2 502= x2+402 2500 = x2+1600 900 = x2 900 = x \30 = x Yes, the lengths are a Pythagorean triple, 30, 40, 50. b) BC2= AB2+ AC2 132= x2+52 169 = x2+25 144 = x2 144 = x \12 = x Yes, the lengths are a Pythagorean triple, 5, 12, 13.

7. Geometry Geometry Theorem 9-2 Converse of The Pythagorean Theorem. Page 245 Page 245 POINTS, LINES AND PLANES 9 9- -1 The Pythagorean Theorem 1 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. Notes Ex: A triangle has side lengths 15, 17, & 20 is it a right triangle? longest2=leg2+leg2 202=152+172 400 = 225+289 400 514 \Not a right D.

8. Geometry Geometry Page 245 Page 245 POINTS, LINES AND PLANES 9 9- -1 The Pythagorean Theorem 1 The Pythagorean Theorem Theorem 9-3 Pythagorean Inequalities Theorem. For any ???, where c is the length of the longest side, the following statements are true. Notes Ex: Verify that the segments form a triangle. Is the triangle acute or obtuse? a) 16, 20, 23 b) 15, 8, 23 16+20 =36 36 > 23 \It's a D. longest2?leg2+leg2 232?162+202 529?256+400 529 < 656 \Acute D. 15+8= 23 23not > 23 \It's not a D.

9. Geometry Geometry Page 246 Page 246 POINTS, LINES AND PLANES 9 9- -1 The Pythagorean Theorem 1 The Pythagorean Theorem In Exercises 1-6, find the value of x. Then tell whether the side lengths form a Pythagorean Triple. 82= x2+42 64 = x2+16 48= x2 48 = x \6.9282 x x2=812 2+1082 x2= 6,561+11,664 x2=18,225 x = 18,225 \x =135 x2=152 2+202 x2= 225+400 x2= 625 x = 625 \x = 25 No, the lengths are not a Pythagorean triple. Yes, the lengths are a Pythagorean triple, 81, 108, 135 or 3(27), 4(27), 5(27). Yes, the lengths are a Pythagorean triple, 15, 20, 25 or 3(5), 4(5), 5(5).

10. Geometry Geometry Page 246 Page 246 POINTS, LINES AND PLANES 9 9- -1 The Pythagorean Theorem 1 The Pythagorean Theorem In Exercises 1-6, find the value of x. Then tell whether the side lengths form a Pythagorean Triple. Do these on your own, I ll give you the answers next class. Check your answers in your table group.

11. Geometry Geometry Page 246 Page 246 POINTS, LINES AND PLANES 9 9- -1 The Pythagorean Theorem 1 The Pythagorean Theorem (Biking south and then east ) = 1.2 + 0.5 School = 1.7 miles 1.2 (Biking diagonal)2 = (1.2)2 +(0.5)2 = 1.44+.25 = SQUARE ROOT 1.69 = 1.3 miles Home 0.5 How many fewer miles would you have biked? 1.7 1.3 = 0.4 miles

12. Geometry Geometry Page 246 Page 246 POINTS, LINES AND PLANES 9 9- -1 The Pythagorean Theorem 1 The Pythagorean Theorem In Exercises 8 and 9, verify that the segment lengths form a triangle. Is the triangle acute, right, or obtuse? 3 1.7320 1+1= 2 2 >1.7320 \It's a D. longest2?leg2+leg2 ( 3)2?12+12 ( 9)?1+1 3> 2 \Obtuse D. 90+216 =306 306 > 234 \It's a D. longest2?leg2+leg2 2342?902+2162 54,756?8,100+46,656 54,756 = 54,756 \Right D.

13. The end KHAN ACADEMY NEXT