M A. 9 1 2 . 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 9.3In 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 TheoremIn 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 c2 = a2 + b2.

9.2 THE PYTHAGOREAN THEOREM

• Homework: Page 538 8-30 even

9.3 THE CONVERSE OF THE PYTHAGOREAN THEOREM

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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 c2 = a2 + b2, 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 c2 = a2 + b2, then ABC is a right triangle. (3,4,5)

a

b

c

A

B

C

9.3 THE CONVERSE OF THE PYTHAGOREAN THEOREM

• Theorem 9.6If 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 c2 < a2 + b2, then ABC is acute.

a

b c

A

BC

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 c2 < a2 + b2, then ABC is acute. (7,12,13)

a

b c

A

BC

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 c2 > a2 + b2, then ABC is obtuse.

a

bc

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 c2 > a2 + b2, then ABC is obtuse. (2,3,5)

a

bc

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

• Theorem 9.8 45-45-90 Triangle TheoremIn a 45-45-90 triangle, the hypotenuse is

times as long as each leg.

x

x

𝑥√2

45

45

9.4 SPECIAL RIGHT TRIANGLES

• Theorem 9.8 45-45-90 Triangle TheoremIn a 45-45-90 triangle, the hypotenuse is

times as long as each leg.

x

x

𝑥√2

45

45

9.4 SPECIAL RIGHT TRIANGLES

• Theorem 9.9 30-60-90 Triangle TheoremIn a 30-60-90 triangle, the hypotenuse is

twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.

2x

x

𝑥√3

60

30

9.4 SPECIAL RIGHT TRIANGLES

• Theorem 9.9 30-60-90 Triangle TheoremIn a 30-60-90 triangle, the hypotenuse is

twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.

2x

x

𝑥√3

60

30

9.4 SPECIAL RIGHT TRIANGLES

• Homework: Page 554 12-30 even

9.5 TRIGONOMETRIC RATIOSMA.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

• Trigonometric Ratios:• Let ABC be a right triangle. The sine, the cosine, and the

tangent of the acute A are defined as follows.

• Sin A = = • Cos A = = • Tan A = =

9.5 TRIGONOMETRIC RATIOS

• Homework: Page 562 10-38 even