1. Today’s Class Do now: – Work on Warm UP – Get out HW Objective – SWBAT apply the Pythagorean theorem to solve for missing side lengths – SWBAT apply the converse of the Pythagorean theorem to classify a triangle by its angles – SWBAT to apply Pythagorean formula to distance problems Homework: Problem Set, Corrections, Binder

2. Announcements Everyone needs a Binder by Friday -Three rings -Only for geometry Retakes due Friday at 5:00pm Quiz Friday

3. We will learn to… Apply the Pythagorean theorem to solve for missing side lengths apply Pythagorean formula to distance problems

4. Background Vocabulary radical This is also called a “Square Root” radicand

5. Steps to Simplify Radicals 1.Find the largest __________________ of the radicand, excluding 1. Note: A factor is a number that divides evenly into a given number with no remainder. 2.Write the number as the _______ of the perfect square factor and its other factor. 3.Split each factor into two separate _______. 4.Simplify the _____________ radical. perfect square factor product radicals perfect square

6. Example 1: 1.Find the largest perfect square factor of 75, excluding 1. 2.Write the radicand as the product of the perfect square factor and its corresponding factor. Check for perfect square factors of 75: 4, 16, 25, 36, 49, 64

7. Example 1: 3.Split each factor into two separate radicals. 4.Simplify the perfect square radical.

8. Example 2:

9. More Simplification… What if the number under the radical is a fraction?

10. Example 3: Rationalizing the Denominator 2. Simplify each radical if possible. Is this radical in simplest form? No, we cannot have a radical in the denominator. 1. Separate the numerator and denominator into separate radicals. We must rationalize the denominator.

11. Example 3: Rationalizing the Denominator 4. Simplify. 3. Multiply both the numerator and denominator by the root. Why can we do this without changing the result?

12. Example 4: Rationalizing the Denominator Simplify the radical. Rationalize the denominator.

13. The values in the chart represent the sides of a right triangle. Complete the chart below. Compare the values of a 2 + b 2 and c 2. Write an algebraic equation to represent this relationship. Describe what the variables in your equation above represent by completing the following phrases: - ∆XYZ is a _____________ triangle because ___________________________________________________________ - a and b represent the _____________ of ∆XYZ because ________________________________________________ - c represents the __________________ of ∆XYZ because ________________________________________________ 9 16 25 144169.36.6411 LEGS they form the right angle of the triangle HYPOTENUSE c 2 = a 2 + b 2 right it has a box (right angle) it is opposite the right angle of the triangle

14. Pythagorean Theorem: If a triangle is a _____________ triangle, then the square of the longest side (______________) is equal to the ____________of the squares of the other two sides (__________). RIGHT HYPOTENUSE SUM LEGS OR

15. How does this diagram represent Pythagorean Theorem?

16. Example 1a Find the value of x. Write your answer in simplest radical form and as a decimal rounded to nearest hundredth. SOLUTION (hypotenuse) 2 = (leg) 2 + (leg) 2 Pythagorean Theorem 5 2 = 3 2 + x 2 25= 9 + x 2 16 = x 2 4 = x Find the positive square root. Substitute. Square. Subtract 9 from both sides

17. Example 1a Find the value of x. Write your answer in simplest radical form and as a decimal rounded to nearest hundredth. SOLUTION Not possible because not a right triangle

18. Example 1c SOLUTION (hypotenuse) 2 = (leg) 2 + (leg) 2 Pythagorean Theorem x 2 = 4 + 12 x 2 = 16 Find the positive square root. Substitute. Square. Add. x = 4 Find the value of x. Write your answer in simplest radical form and as a decimal rounded to nearest hundredth. 2

19. Example 1c Maritza and Melanie run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest hundredth of a mile, they must travel to return to their starting point?

20. EXAMPLE 4 SOLUTION =+ 16 2 = 4 2 + x 2

21. EXAMPLE 3 Standardized Test Practice Find positive square root. Substitute. Square. Subtract 16 from each side. SOLUTION Approximate with a calculator. 16 2 = 4 2 + x 2 256 = 16 + x 2 15.491 ≈ x 240 = x 240 = x 2 ANSWER The ladder is resting against the house at about 15.5 feet above the ground. The correct answer is D.

22. Example 4 SOLUTION (hypotenuse) 2 ? (leg) 2 + (leg) 2 Pythagorean Theorem 64 ?16 + 48 64 ?64 YES, these side lengths make a right triangle! Substitute. Square Add. 64 = 64

23. Distance Formula and Pythagorean Theorem Distance Formula: Pythagorean formula:

24. Distance Formula and Pythagorean Theorem Distance Formula: Pythagorean formula:

25. We will learn to… Apply the converse of the Pythagorean Theorem to classify a triangle by its angles.

26. Using Pythagorean Converse Activity Work with a partner Each pair should have a bag with pieces of spaghetti of different lengths – Make sure to measure the pieces to ensure that you are using the correct pieces for each trial Fill in the table for each trial Answer the ‘Analysis’ questions that follow

27. Is that a triangle?? NO! YES!

28. Yes: 7 > 59 16 25 right Yes: 9 > 6 16 25 36 41 acute Yes: 5 > 4 4 916 13 obtuse Yes – For all trials, a + b > c No, c 2 = a 2 + b 2 is only true for right triangles. < = > Used to classify a triangle as right, obtuse or acute

29. Example 1 Can segments with lengths of 6.1 feet, 5.2 feet, and 4.3 feet form a triangle? If so, classify the triangle by its angles.

30. Example 2 Determine whether segments with lengths 3, 8, and 7 can form a triangle. If they can, classify the triangle by angles.

31. Example 3 Classify each triangle below by its angles and sides.

32. Pythagorean Triples: Any set of _____ ____________ numbers {a, b, c} that satisfies _______________.  IF it satisfies the rule stated above, then the three numbers create a _______ ____________. Example 5: Determine if the following sets of numbers are Pythagorean Triples. Justify your response. three whole c 2 = a 2 + b 2 right triangle

33. Example 5: Determine if the following sets of numbers are Pythagorean Triples. Justify your response. b. {7, 9, 8}c. {37, 12, 35} c 2 ? a 2 + b 2 9 2 ? 7 2 + 8 2 81 ? 49 + 64 81 < 113 Since c 2 < a 2 + b 2 then this set of numbers is NOT a Pythagorean Triple c 2 ? a 2 + b 2 37 2 ? 12 2 + 35 2 1369 ? 144 + 1225 1369 = 1369 Since c 2 = a 2 + b 2 then this set of numbers is a Pythagorean Triple