1. Lesson 14.2
Special Triangles pp

2. Objectives:
1. To develop theorems to give the relationships between the sides and hypotenuse of and right triangles.
2. To establish the trig ratios for the and right triangles.
3. To find missing sides or angles of specific and right triangles.

3. The first special triangle is the 45-45 right triangle.
45°
x2 + x2 = c2
2x2 = c2
x 2
c = x 2 x x

4. Theorem 14.1
If the length of a leg of an isosceles right triangle (45-45 right triangle) is x, then the length of the hypotenuse is x 2.

5. Find the three trigonometric ratios for a 45° angle and the lengths of the sides of the triangle.B C
45° 4

6. The second special triangle is the 30-60 right triangle.
30°
60° 2x x

7. The second special triangle is the 30-60 right triangle.
30°
60°
x2 + b2 = (2x)2
x2 + b2 = 4x2
b2 = 3x2 2x
b = x 3 x

8. Theorem 14.2
If the length of the leg opposite the 30° angle of a right triangle is x, then the length of the leg opposite the 60° angle is x 3, and the length of the hypotenuse is 2x.

9. Find the side lengths and trigonometric ratios for the acute angles of the following 30-60
right triangle. X Y Z
60°
30° 10
z =
y =
sin X = sin Y =
cos X = cos Y =
tan X = tan Y =

10. The and right triangles are special because the side relations can be determined using only the Pythagorean theorem.

11. Given the following measures of the sides of a right triangle, determine if the triangle is 45-45, 30-60, or neither.
5, 5, 30
3. Neither

12. Given the following measures of the sides of a right triangle, determine if the triangle is 45-45, 30-60, or neither.
6, 2 3, 4 3
3. Neither

13. Given the following measures of the sides of a right triangle, determine if the triangle is 45-45, 30-60, or neither.
4, 4, 4 2
3. Neither

14. Given the following measures of the sides of a right triangle, determine if the triangle is 45-45, 30-60, or neither.
9, 12, 15
3. Neither

15. Homework pp

16. ►A. Exercises A B C
45° X Y Z
60°
30°
For each of the given measures, find the length of the sides of the triangles.
AB AC BC
units

17. ►A. Exercises A B C
45° X Y Z
60°
30°
For each of the given measures, find the length of the sides of the triangles.
AB AC BC
units

18. ►A. Exercises A B C
45° X Y Z
60°
30°
For each of the given measures, find the length of the sides of the triangles.
XY XZ YZ
7. 12 units

19. ►A. Exercises A B C
45° X Y Z
60°
30°
For each of the given measures, find the length of the sides of the triangles.
XY XZ YZ
units

20. ►B. Exercises
Given the following side lengths, determine which of these triangles are special and give the name of each special right triangle.
11. 6, 6, 6 2

21. ►B. Exercises
Given the following side lengths, determine which of these triangles are special and give the name of each special right triangle.
13. 7, 7 3, 14

22. ►B. Exercises
Given the following side lengths, determine which of these triangles are special and give the name of each special right triangle.
15. 12, 12,

23. ►B. Exercises
Give the following trigonometric ratios. Draw pictures if necessary.
17. sin 45°

24. ►B. Exercises
Give the following trigonometric ratios. Draw pictures if necessary.
19. tan 60°

25. ►B. Exercises
Give the following trigonometric ratios. Draw pictures if necessary.
21. cos 60°

26. ■ Cumulative Review
Give the dimensions of the figure with the following areas.
24. A square with area of 20 square units

27. ■ Cumulative Review
Give the dimensions of the figure with the following areas.
25. An equilateral triangle with an area of square units.

28. ■ Cumulative Review
Give the dimensions of the figure with the following areas.
26. A regular hexagon with an area of
50 3 square units

29. ■ Cumulative Review
Give the dimensions of the figure with the following areas.
27. A circle with an area of 5 square units

30. ■ Cumulative Review
Give the dimensions of the figure with the following areas.
28. A cube with a surface area of square units 8 3