Bellwork

1. Clickers Bellwork A B Find m<DBC in square ABCD In the previous example, find the area of triangle BDC What is the length of segment BD What is the area of an equilateral triangle with side length 10 What is the altitude(height) of an equilateral triangle with side length of 20 What is the altitude(height) of an equilateral triangle with side length of 32 x C D

2. Bellwork Solution A B Find m<DBC in square ABCD In the previous example, find the area of triangle BDC What is the length of segment BD x C D

3. Bellwork Solution What is the area of an equilateral triangle with side length 10 What is the altitude(height) of an equilateral triangle with side length of 20 What is the altitude(height) of an equilateral triangle with side length of 32

4. Special Right Triangles Section 7.4

5. The Concept • Today we’re going to be working with some special right triangles that occur within other geometric figures • The ratios shown today will reappear throughout your odyssey throughout mathematics…

6. Special Triangle There are two special right triangles that occur naturally and are thus often studied in trigonometry. We’re going to look at one today and the other on Monday What’s the length of the diagonal of a square of side length 1 1 This kind of isosceles triangle is called a 45-45-90 triangle because of the angles formed when you draw a diagonal across a square 1

7. Special Triangle How would this relate to triangles whose sides are larger than 1? 5 Therefore the relationship between sides and hypotenuse this triangle is by radical two… 5

8. The theorem Theorem 7.8 In a 45-45-90 triangle, the hypotenuse is radical 2 times as long as each leg

9. On your own Solve for x

10. On your own Solve for x

11. On your own Solve for x

12. On your own Solve for x

13. Special Triangle Given an equilateral triangle of side length 2, can we determine the height of the triangle 2 h This kind of triangle is called a 30-60-90 triangle because of the angles formed when you draw an altitude in an equilateral triangle 1 2

14. Special Triangle How would this relate to triangles whose sides are larger than 1? 6 h Therefore the relationship the 60o side and hypotenuse is one half and the other side is by radical three 3 6

15. The theorem Theorem 7.9 In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. 30o 60o

16. On your own Solve for x

17. On your own Solve for x

18. On your own Solve for x

19. On your own Solve for x & y

20. On your own Solve for x

21. On your own Solve for x

22. On your own Solve for x & y

23. Homework 7.4 1, 2-20 even, 23-25, 27-33 odd, 28

24. Practical example The distance from Bill’s feet to his waist is 3 feet. While doing leg lifts at track practice, he wonders how high his feet are off the ground in order to get his mind off the burning pain in his abs. How far off the ground are his feet when his legs make a 30o angle with the ground? • 1.5 ft • 2.60 ft • 5.20 ft

25. Practical example The distance from Bill’s feet to his waist is 3 feet. While doing leg lifts at track practice, he wonders how high his feet are off the ground in order to get his mind off the burning pain in his abs. How far off the ground are his feet when his legs make a 45o angle with the ground? • 2.12 ft • 3 ft • 4.24 ft

26. Practical example The distance from Bill’s feet to his waist is 3 feet. While doing leg lifts at track practice, he wonders how high his feet are off the ground in order to get his mind off the burning pain in his abs. How far off the ground are his feet when his legs make a 60o angle with the ground? • 1.5 ft • 2.60 ft • 5.20 ft

27. Most Important Points • 45-45-90 triangle side relationships • 30-60-90 triangle side relationships