 # MA.912.T.2.1 CHAPTER 9: RIGHT TRIANGLES AND TRIGONOMETRY.

## Presentation on theme: "MA.912.T.2.1 CHAPTER 9: RIGHT TRIANGLES AND TRIGONOMETRY."— Presentation transcript:

MA.912.T.2.1 CHAPTER 9: RIGHT TRIANGLES AND TRIGONOMETRY

9.1 SIMILAR RIGHT TRIANGLES MA.912.T.2.1

9.1 SIMILAR RIGHT TRIANGLES Theorem 9.1 IF the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

9.1 SIMILAR RIGHT TRIANGLES Theorem 9.2 In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments.

9.1 SIMILAR RIGHT TRIANGLES Theorem 9.3 In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of they hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

9.1 SIMILAR RIGHT TRIANGLES Homework: Page 531 14-30 Even

9.2 THE PYTHAGOREAN THEOREM MA.912.T.2.1

9.2 THE PYTHAGOREAN THEOREM Theorem 9.4 Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Pythagorean Triple – A set of three positive integers, a, b, and c, that satisfy the equation c 2 = a 2 + b 2.

9.2 THE PYTHAGOREAN THEOREM Homework: Page 538 8-30 even

9.3 THE CONVERSE OF THE PYTHAGOREAN THEOREM MA.912.T.2.1

9.3 THE CONVERSE OF THE PYTHAGOREAN THEOREM Theorem 9.5 Converse of the Pythagorean Theorem If the square of the length of the sum of the side of a triangle is equal to the sum of the two squares of the lengths of the other two sides, then the triangle is a right triangle. IF c 2 = a 2 + b 2, then ABC is a right triangle. a b c A B C

9.3 THE CONVERSE OF THE PYTHAGOREAN THEOREM Theorem 9.5 Converse of the Pythagorean Theorem Paraphrase: If the sides of a triangle can be substituted into the Pythagorean theorem and simplify to a true statement, then the triangle is a right triangle. IF c 2 = a 2 + b 2, then ABC is a right triangle. (3,4,5) a b c A B C

9.3 THE CONVERSE OF THE PYTHAGOREAN THEOREM Theorem 9.6 If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute. IF c 2 < a 2 + b 2, then ABC is acute. a b c A B C

9.3 THE CONVERSE OF THE PYTHAGOREAN THEOREM Theorem 9.6 Paraphrase: If the sides of a triangle can be substituted into the Pythagorean theorem and simplify to the longest side squared is smaller than the sum of the legs squared, then the triangle is an acute triangle. IF c 2 < a 2 + b 2, then ABC is acute. (7,12,13) a b c A B C

9.3 THE CONVERSE OF THE PYTHAGOREAN THEOREM Theorem 9.7 IF the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse. IF c 2 > a 2 + b 2, then ABC is obtuse. a b c A B C

9.3 THE CONVERSE OF THE PYTHAGOREAN THEOREM Theorem 9.7 Paraphrase: If the sides of a triangle can be substituted into the Pythagorean theorem and simplify to the longest side squared is greater than the sum of the legs squared, then the triangle is an obtuse triangle. IF c 2 > a 2 + b 2, then ABC is obtuse. (2,3,5) a b c A B C

9.3 THE CONVERSE OF THE PYTHAGOREAN THEOREM Homework: Page 546 8-24 even

9.4 SPECIAL RIGHT TRIANGLES MA.912.T.2.1

9.4 SPECIAL RIGHT TRIANGLES x x 45

9.4 SPECIAL RIGHT TRIANGLES x x 45

9.4 SPECIAL RIGHT TRIANGLES 2x x 60 30

9.4 SPECIAL RIGHT TRIANGLES 2x x 60 30

9.4 SPECIAL RIGHT TRIANGLES Homework: Page 554 12-30 even

9.5 TRIGONOMETRIC RATIOS MA.912.T.2.1

9.5 TRIGONOMETRIC RATIOS Trigonometric Ratio – a ratio of lengths of two sides of a right triangle. Sine, cosine, and tangent Angle of elevation – the angle that your line of sight makes with a line drawn horizontally.

9.5 TRIGONOMETRIC RATIOS

Homework: Page 562 10-38 even