1. Pythagorean Theorem إعداد الطالب : إعداد الطالب : شهاب فؤاد عبد المنعم صبرى شهاب فؤاد عبد المنعم صبرى الصف الأول الإعدادى الصف الأول الإعدادى تحت إشراف : تحت إشراف : أ / تامر أ / تامر

2. Pythagorean Theorem Objectives: 1) To use the Pythagorean Thm. 2) To use the converse of the Pythagorean Thm.

3. 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.

4. 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.

6. 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)

7. Pythagorean Theorem a 2 + b 2 = c 2 Legs Hyp Hypotenuse. – Longest side of a rt. Δ ** Only works for rt. Δs c a b Legs * Sides that form the right 

8. 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 = x

9. 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 …

10. Math Review Perfect Squares –1–1–1–12 = 1 –2–2–2–22 = 4 –3–3–3–32 = 9 –4–4–4–42 = 16 –5–5–5–52 = 25 –6–6–6–62 = 36 –7–7–7–72 = 49 –8–8–8–82 = 64 –9–9–9–92 = 81 Radicals √3 √3 = √9 = 3 √40 = √4 10 = 2√10 √80 = √420 = 2√45 = 22√5 = 4√5

11. 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)

12. Example 3: Find the area of ΔDCE 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 = 6.6m

13. Example 4: Are the following Δs, 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!

14. Non-Right Δs 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.

15. 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.

16. 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 Acute

17. Ex. 7 Find AC and BC The area of ΔABC 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 = √ 45 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 8ft2ft

18. Day 2 page 420, 18-29, 36,39,40,44 Day 1 page 420, 1-17 all