Investigation 4.2

1. Investigation 4.2 AMSTI Searching for Pythagoras

2. Problem of the Day • Using the regular hexagon, find the total number of equal triangles inside. Draw the triangles from point A. How many triangles did you find? • How many degrees are in the hexagon? • How did you find this answer? •A

3. Problem of the Day • There are six triangles in a regular hexagon. • Each triangle is 180°. • The total number of degrees in a regular hexagon is 720°. • You can use the formula to find the answer! (n – 2)180° •A

5. Problem 4.2 (Labsheet 4.2) • Each side of equilateral triangle ABC has a length of 2. • Remember, all sides are equal and all angles are 60°. A 2 2 C B 2

6. Labsheet 4.2 • On labsheet 4.2, find the point halfway between vertices B and C. Label this point P. • Point P is the midpoint of segment BC. • Draw a segment from vertex A to point P. • This divides triangle ABC into two congruent triangles. A C B P

7. Labsheet 4.2 A • Cut out triangle ABC and fold it along line AP. • What do you notice about the two new triangles? C B P

8. Problem 4.2 (A) • How does triangle ABP compare with triangle ACP? Problem 4.2 (B) • Find the measure of each angle in triangle ABP. Explain how you found each measure.

9. Problem 4.2 (C) • Find the length of each side of triangle ABP. Explain how you found each length. Problem 4.2 (D) • Two line segments that meet at right angles are called perpendicular line segments. Find a pair of perpendicular line segments in the drawing.

10. Paper Folding Fold the corners and draw a line. Shade in this area. Fold the corners and draw a line. Shade in this area. Fold your paper up the middle and draw a line.

11. Paper Folding 60° 60° You should have lines like these drawn on your envelope. 30° 30° Trace the two other folds so that you have two more lines like these! 60° 60° 90° 60° 60° 60° 60° 90° What kind of triangles do you see?

12. Problem 4.2 Follow-Up (1) • A right triangle with a 60° angle is sometimes called a 30-60-90 triangle. This 30-60-90 triangle has a hypotenuse of length 6. • What are the lengths of the other two sides? • Explain how you found your answers. 30° 6 60°

13. Problem 4.2 Follow-Up (2) A B • Square ABCD has sides of length 1. • On Labsheet 4.2, draw a diagonal, dividing the square into two triangles. • Cut out the square and fold it along the diagonal. D C

14. Problem 4.2 Follow-Up (2) A B • How do the two triangles compare? • What are the measures of the angles of one of the triangles? Explain. • What is the length of the diagonal? Explain how you found the length. • Suppose square ABCD had sides of length 5. How would this change your answers to parts b and c? D C

15. ACE Questions for 4.2 Answer the following ACE questions on page 47 - #2, 4, 7, and 10

16. Investigation 4.3 AMSTI Searching for Pythagoras

17. Investigation 4.3Special TrianglesFor advanced students or extra enrichment

18. 4.3 Special Triangles • With an equilateral triangle: • All sides are equal (shown by “a”) • All angles are equal (60°) • The perpendicular bisector is the height (h) and it creates two 30-60-90 triangles!

19. 4.3 Special Triangles (30-60-90) • By using a perpendicular bisector, the shorter leg is half of the hypotenuse. (a/2) • Use this with the Pythagorean Theorem to find missing lengths!

20. 4.3 Special Triangles • This right triangle is a 45-45-90. • The two legs are equal (a). • The hypotenuse (h) can be found by using the formula h/√a or the Pythagorean Theorem.

21. Problem (pg 45) A • If length CD is 8, what is the length of AC? • Now, find the length of AD. What length did you find? • What is the length of BC? • What is the length of AB? 60° 30° B C 8 D

22. ACE Problems #8, 9, 11, 12