1. The Special Right Triangle Show Featuring The 45°-45°-90° Triangle And The 30°-60°-90° Triangle

2. Act 1 – Scene 1 Andy 45°-45°-90°

3. Setting Andy 45°-45°-90° Triangle enters the stage and stares into the mirror. As he looks at the image looking back at him he notices something he has never noticed before.

4. The Look Wow, I have never noticed that the sides of my face are the SAME LENGTH!!

5. Making Connections It’s kind of like they have a relationship and I can call them both x. X X

6. Closer Look Andy 45°-45°-90° stared closely into the mirror and noticed something else. He could not believe his eyes. He thought to himself this must be the same for all of my 45°-45°-90° brothers and sisters, cousins, all of the 45°-45°-90° clan!

7. *Bright Idea* XX 45° Both the X’s are opposite my 45° angles.

8. *Even Brighter Idea* XX 45° The length of the bottom of my face is

9. The Moment of Truth Andy 45°-45°-90° started to think to himself and realized that in all 45°- 45°-90° triangles, whatever value is chosen for x, it remains the same for all the x’s. All 3 sides of his face had a relationship with each other! When he knew the length of one, he could find the missing sides!

10. A Second Glance XX 45° I measured the side of my face and it is 6 inches! That must mean both sides of my face are 6 inches. 6 6 That also must mean that the bottom of my face is ….

11. The Family Reunion The 45°-45°-90°’s always planned a family reunion about this time of year. Nice time to get together and celebrate the holidays. Andy 45°- 45°-90° was ready to wow his cousins with his new found information. He was going to ask his cousin for the length of one side of his face and tell him he could find the other two sides!

12. Presenting Alex 45°-45°-90° The Great! Jordan, bettcha if you give me the length of one side of your face, I can tell you the other two! NO WAY! Prove it! The length of the left side of my face is 3. 3

13. The Challenge Ha so easy! The right side of your face is 3 also and the bottom of your face isWOW Andy, your amazing!

14. A Closer Look 45° X X All three X’s are the SAME VALUE!

15. Example A Given a 45°-45°-90° triangle, with the measure of AB = 12. What are the other two sides? 45° 12 C A B 12 AC = BC =

16. Example B Given a 45°-45°-90° triangle, with the measure of BC = 16. What are the other two sides? 16 C A B AB = BC =

17. How’d We Do That?! 16 C A B We know that 16 must be = LET’S SOLVE FOR X! Set equal to each other Divided both sides by radical 2, to get x alone. Multiply left side by radical 2 over radical 2.

18. How’d We Do That Cont. Radical 2 times radical 2 simply leaves us with 2. Reduce 16 over 2 to 8. RULE When given the hypotenuse of a 4 44 45°-45°-90° triangle, simply divide it by 2 and multiply it by radical 2 for the length of the other two sides.

19. IntermissionTake this time to work on a few examples of 45°-45°-90° triangles! Remember the rules and what you know about the lengths of the sides. 45° X X

20. Act 1 – Scene 2 Meeting His Match

21. A Show On The Road Andy then took his show on the road traveling from city to city showing that he could determine the missing lengths of the sides of a 45°-45°-90° when given just a single side.

22. The Show My hypotenuse is equal to 14 what are my sides?! Seth, that’s a joke right? SO EASY, both of your sides are 7√2 Gosh! You got it!

23. The Show One of my sides equals 6. What is the other one and the hypotenuse? David, that one is even easier than the one before! The other side is 6 and your hypotenuse is 6√2 How’d you do that!!!

24. I think I can stump you Andy, my side equals 5√2. Nice try Kayla! But I have figured you out as well, your other side is 5√2. And, Miss your hypotenuse is 10! WOW he is amazing!

25. You’ll never figure me out! One of my sides equals 12√3! Once again, Jada, I have figured you out as well, your other side is 12√3. your hypotenuse is 12√6! Simply Amazing, Ms. Balboni must have taught him!

26. This is certainly the one to stump you! One of my sides equals 5√8! If you thought you got me, you haven’t, your other side is 5√8. And, Thomas your hypotenuse is 20! I give up, he really knows his stuff!

27. How about me Andy! On of my sides is 5!! Andy look and he stared. He had never seen anything quite like Robin. He knew she wasn’t a 45°- 45°-90° but he did know there was something special about her! Come on Andy. Don’t tell me I stumped you!

28. Act 2 – Scene 2 The Start of Something New

29. At First Sight At First Sight You’re different Robin, I have never met anyone like you.You’rejust saying that Andy!

30. No really! You are! All of your sides are DIFFERENT lengths!!!! O…

31. No, its really cool, I think I can determine a relationship between all 3 of them!!. A relationship? What are you talking about?

32. What I mean is a set of rules so when I know one side, I can figure out the others. Andy, you’re so smart!

33. A Closer Look A Closer LookWell Robin, it seems to me that your hypotenuse is twice the length of the side opposite the 30° angle. I shall call them y and 2y. y 2y 30° 30° 60°

34. A Closer Look A Closer LookNow that third side, that’s a little tricky, but I think I got it! If you use the measure opposite the 30° angle and multiply it by √3. IT WORKS!!! y√3 2y 30° 30° 60° y

35. Amazed Amazed Ha there you Robin! Those are the relationships between your sides! Andy, wow I am truly amazed!

36. Taking it all in Taking it all in y 2y y√3 Lets make the side opposite the 30° equal to 5. Let’s see if we can figure out the other two sides. 5 10 5√3

37. Example 2 Example 2 y y√3 Given the hypotenuse is equal to 12, determine the length of the other two sides. 6 12 6√3

38. Example 3 Example 3 Lets make the side opposite the 60° equal to 9. Let’s see if we can figure out the other two sides. 9 3√3 6√3 6√3

39. Act 2 – Scene 3 Simply the Best

40. Rule When you are given the measure of the side opposite the 60° angle, to find y or the measure opposite the 30° angle, you divide it by 3 and then multiply it by radical 3. 15 60° == Which makes the side opposite the 90° angle equal to….

41. Love is in the Air Robin was so impressed by Andy that they began to spend a lot of time together. She would go with him on tour while he now determined the sides of 30°-60°-90° and 45°-45°-90° triangles. She was constantly impressed by his ability.

42. World Tour My hypotenuse is equal to 12 what are my sides?! He shoots! He scores!! Simple, easy as pie Gianny, the side opposite the 30° angle is 6 and the side opposite the 60° is 6√3 60° 30°

43. World Tour Try this one Andy, side opposite my 30° angle is 6√3! Wow, I thought that would stump you Nice try Joey, but I am sure your hypotenuse is 12√3 and opposite the 60° is 18! 60° 30°

44. World Tour Try this one Andy, side opposite my 30° angle is 8√3! Right again! Good JOB! Sorry Jayla, I’ve figured you out too your hypotenuse is 16√3 and opposite the 60° is 24! 60° 30°

45. World Tour How about this one my hypotenuse is 4√3! Geez, you are good! I’m amazed! Oh tricky tricky Jaila, but I have you figured out as well. Opposite the 30° is 2√3 and opposite the 60° is 6! 60° 30°

46. World Tour How about the side opposite the 60° is 6√5! Impressive! Yup, got you as well, can anyone give me challenge? Veronica, the side opposite the 30° is 2√15 and your hypotenuse is 4√15! 60° 30°

47. World Tour I think I got one to stump you! The side opposite my 60° angle is 7! Ms. Balboni must be the best teacher! Hahahah Oh Olivia, you are foolish, of course I can figure this one out. The side opposite the 30° is 7√3 3 and your hypotenuse is 14√3 3 60° 30°

48. The Show One of my sides equals = 5 What is the other one and the hypotenuse? Zach, that one is even easier than the one before! The other side is 5 and your hypotenuse is 5√2 Erin, you’re a lucky lady

49. World Tour Try this one Andy, side opposite my 30° angle is 9√3! You’re goooood! Nice try Korina I am sure your hypotenuse is 18√3 and opposite the 60° is 27! 60° 30°

50. The Show We’re congruent and one of our sides equals = 12.5√2 What is the other one and the hypotenuse? Jade and Alicia, nice try but the other three sides are 12.5√2 and both your hypotenuses are 25. Ha you figured us out too

51. World Tour Ok, this might stump you. The measure of the angle opposite the 30° is 10√3! Oh man, I thought for sure I would get you! Sorry Taylor, but you haven’t fooled me yet! Your hypotenuse is 20√3 and opposite the 60° is 30 ! 60° 30°

52. World Tour How about this one my hypotenuse is 3√6! Wow, that’s amazing!! Good one Luis, but still very simple for me! well. Opposite the 30° is 1.5√6 and opposite the 60° is 4.5√2! 60° 30°

53. Scene 3 – Act 1 And So It Went

54. Making Connections And so it went, Andy toured around the world, amazing people day by day, but it began to get old, people were getting bored. He had to come up with something else, something that would amaze the crowds! Then he yelled, “ I GOT IT!”

55. Making Connections 2 The measures of equilateral triangle can also be determined due to the use of special right triangles.