1. Special Right Triangles
Geometry 7-3

2. Special Right Triangles 45 – 45 - 90
Geometry 7-3a

3. Review

4. Areas

5. Area of a Triangle
The area of a triangle is given by the formula A = ½ B x H, where A is the area, B is the length of the base, and H is the height of the triangle
Area

6. Theorem The Pythagorean theorem
In a right triangle, the sum of the squares of the legs of the triangle equals the square of the hypotenuse of the triangle B c a A C b
Theorem

7. Theorem Converse of 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. B c a A C b
Theorem

8. Converse of Pythagorean

9. New Material

10. Copy the following chart into your notes
Investigation

11. Find the length of the hypotenuse of each isosceles right triangle
Find the length of the hypotenuse of each isosceles right triangle. Simplify each square root.
Record the answers in your chart
Investigation

12. Finish the chart for each of the listed leg lengths
Investigation

13. Theorem 45° – 45° – 90° Triangle
In a 45° – 45° – 90° triangle the hypotenuse is the square root of two times as long as each leg
Theorem

14. Question
When the problem says this, How do we reduce the square root of two?
Answer
We don’t, unless it is in the denominator.

15. Know the basic triangle
Set known information equal to the corresponding part of the basic triangle
Solve for the other sides

16. Example

17. Practice

18. Example

19. Practice

20. Practice

21. Practice

22. Practice

23. Practice

24. Special Right Triangles 30 – 60 - 90
Geometry 7-3b

25. Special Triangle Investigation
Draw This in your notes
A large equilateral triangle
Special Triangle Investigation

26. Special Triangle Investigation
Divide the triangle in half
You now have a 30° – 60° – 90° triangle
Special Triangle Investigation

27. Special Triangle Investigation
Label the triangle.
Special Triangle Investigation

28. Special Triangle Investigation
Is AC = BC?
Why?
Yes, Definition of isosceles triangle, or equilateral
Special Triangle Investigation

29. Special Triangle Investigation
Are the two separate triangles congruent?
Why?
Yes, ASA
Special Triangle Investigation

30. Special Triangle Investigation
Is AD = BD?
Why?
Yes, CPCTC
Special Triangle Investigation

31. Special Triangle Investigation
So, AC = AB, and AD = DB;
What is the relationship between AC and AD?
AC = 2 x AD
Special Triangle Investigation

32. Special Triangle Investigation
So AC = 2 AD
Using the Pythagorean theorem, what is the length of CD, in terms of AD?
Special Triangle Investigation

33. Theorem 30° – 60° – 90° Triangle
In a 30° – 60° – 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is the square root of three times as long as the shorter leg
Theorem

34. Solving Strategy Know the basic triangles
Set known information equal to the corresponding part of the basic triangle
Solve for the other sides
Solving Strategy

35. Know the basic triangles
Set known information equal to the corresponding part of the basic triangle
Solve for the other sides

36. Example

37. Example

38. Practice

39. Practice

40. Practice

41. Practice

42. Practice

43. Practice

44. Practice

45. Practice

46. Pages 369 – 372
2 – 8 even, 12 – 28 even, 34 – 38 even, 47, 48
Homework