Lesson 14.2 Special Triangles pp. 588-591

Objectives: 1. To develop theorems to give the relationships between the sides and hypotenuse of 45-45 and 30-60 right triangles. 2. To establish the trig ratios for the 45-45 and 30-60 right triangles. 3. To find missing sides or angles of specific 45-45 and 30-60 right triangles.

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

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.

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

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

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

Theorem 14.2 If the length of the leg opposite the 30° angle of a 30-60 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.

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 =

The 45-45 and 30-60 right triangles are special because the side relations can be determined using only the Pythagorean theorem.

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 1. 45-45 2. 30-60 3. Neither

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 1. 45-45 2. 30-60 3. Neither

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 1. 45-45 2. 30-60 3. Neither

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 1. 45-45 2. 30-60 3. Neither

Homework pp. 590-591

►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 1. 18 units

►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 5. 4 units

►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

►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 9. 2 units

►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

►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

►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, 12 3

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

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

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

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

■ Cumulative Review Give the dimensions of the figure with the following areas. 25. An equilateral triangle with an area of 100 3 square units.

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

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

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