Right Triangles and Trigonometry Chapter 9

The Pythagorean Theorem and Its Converse I can use the Pythagorean Theorem and the Converse of the Pythagorean Theorem.

The Pythagorean Theorem and Its Converse Vocabulary (page 244 in Student Journal) Pythagorean triple: a set of nonzero whole numbers that satisfy the equation a2 + b2 = c2

The Pythagorean Theorem and Its Converse Core Concepts (pages 244 and 245 in Student Journal) Pythagorean Theorem If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b2 = c2

The Pythagorean Theorem and Its Converse Converse of the Pythagorean Theorem If the sum of the squares of the length of 2 sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle

The Pythagorean Theorem and Its Converse Pythagorean Inequalities Theorem 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 2 sides, then the triangle is obtuse. less than the sum of the squares of the lengths of the other 2 sides, then the triangle is acute.

The Pythagorean Theorem and Its Converse Examples (page 246 in Student Journal) Find the value of x. Do the side lengths form a Pythagorean triple? #2) #3)

The Pythagorean Theorem and Its Converse Solutions #2) 4√3, no #3) 25, yes

The Pythagorean Theorem and Its Converse Verify the segment lengths form a triangle. Is the triangle acute, right, or obtuse? #8) 90, 216, 234

The Pythagorean Theorem and Its Converse Solution #8) 90 + 214 > 234, right because 90 2 + 214 2 = 234 2

Special Right Triangles I can find side lengths in special right triangles.

Special Right Triangles Core Concepts (page 249 in Student Journal) 45-45-90 Triangle Theorem In a 45-45-90 triangle, both legs are congruent and the length of the hypotenuse is the square root of 2 times the length of a leg.

Special Right Triangles 30-60-90 Triangle Theorem In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg and the length of the longer leg is the square root of 3 times the length of the shorter leg.

Special Right Triangles Examples (page 250 in Student Journal) Find the value of x. Write your answer in simplest form. #1) #3)

Special Right Triangles Solutions #1) 10√2 #3) 8

Special Right Triangles #6) Find the value of x and y. Write the answer in simplest form.

Special Right Triangles Solution #6) x = 11√3, y = 11

Similar Right Triangles I can use geometric means.

Similar Right Triangles Vocabulary (page 254 in Student Journal) geometric mean: the positive number x that satisfies the equation a/x = x/b, where a and b are positive numbers

Similar Right Triangles Core Concepts (pages 254 and 255 in Student Journal) Right Triangle Similarity Theorem If the altitude is drawn to the hypotenuse of a right triangle, then the 2 triangles formed are similar to the original triangle and to each other.

Similar Right Triangles Geometric Mean (Altitude) Theorem In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into 2 segments. The length of the altitude is the geometric mean of the lengths of the 2 segments of the hypotenuse.

Similar Right Triangles Geometric Mean (Leg) Theorem In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into 2 segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

Similar Right Triangles Examples (page 256 in Student Journal) #1) Identify the similar triangles in the diagram.

Similar Right Triangles Solution #1) triangle IJH ~ triangle IKJ ~ triangle JKH

Similar Right Triangles Find the value of the variable. #5) #6)

Similar Right Triangles Solutions #5) x = 12 #6) y = 3√11

The Tangent Ratio I can use the tangent ratio.

The Tangent Ratio Vocabulary (page 259 in Student Journal) trigonometric ratio: equivalent ratios created from the corresponding sides of similar right triangles tangent (tan): the ratio of the length of the opposite leg to the length of the adjacent leg

The Tangent Ratio angle of elevation: the angle above a horizontal line

The Tangent Ratio Examples (pages 260 and 261 in Student Journal) #1) Find the tangents of the acute angles.

The Tangent Ratio Solution #1) tan <R = 15/8, tan <S = 8/15

The Tangent Ratio #4) Find the value of x.

The Tangent Ratio Solution #4) x = 28.4

The Tangent Ratio #9) A boy flies a kite at an angle of elevation of 18 degrees. The kite reaches its maximum height 300 feet away from the boy. What is the maximum height of the kite?

The Tangent Ratio Solution #9) 97.5 feet

The Sine and Cosine Ratios I can use the sine and cosine ratios.

The Sine and Cosine Ratios Vocabulary (page 264 in Student Journal) sine (sin): the ratio of the length of the opposite leg to the length of the hypotenuse cosine (cos): the ratio of the length of the adjacent leg to the length of the hypotenuse

The Sine and Cosine Ratios angle of depression: the angle below a horizontal line

The Sine and Cosine Ratios Core Concepts (pages 264 and 265 in Student Journal) SOH-CAH-TOA Sine and Cosine of Complementary Angles The sine of an acute angle is equal to the cosine of its complement and vice versa.

The Sine and Cosine Ratios Examples (pages 265 and 266 in Student Journal) #1) Find sin F, sin G, cos F, cos G.

The Sine and Cosine Ratios Solution #1) sin F = 12/13, sin G = 5/13 cos F = 5/13, cos G = 12/13

The Sine and Cosine Ratios Find the value of each variable. #10)

The Sine and Cosine Ratios Solution #10) x = 2.5, y = 8.7

Solving Right Triangles I can use inverse trigonometric ratios to solve right triangles.

Solving Right Triangles Vocabulary (page 269 in Student Journal) inverse tangent (tan-1): if tan A = x, then tan-1x = measure of angle A inverse sine (sin-1): if sin A = y, then sin-1y = measure of angle A

Solving Right Triangles inverse cosine (cos-1): if cos A = z, then cos-1z = measure of angle A solve a right triangle: finding all unknown side lengths and angle measures in a right triangle

Solving Right Triangles Examples (pages 270 and 271 in Student Journal) #7) Solve the right triangle.

Solving Right Triangles Solution #7) AC = 13.4, m<A = 63.4 degrees, m<C = 26.6 degrees

Solving Right Triangles #11) A boat is pulled by a winch on a dock 12 feet above the dock of the boat. When the winch is fully extended to 25 feet, what is the angle of elevation from the boat to the winch?

Solving Right Triangles Solution #11) 28.7 degrees

Law of Sines and Law of Cosines I can use the Law of Sines and the Law of Cosines to solve triangles.

Law of Sines and Law of Cosines Vocabulary (page 274 in Student Journal) Law of Sines: used to solve triangles when 2 angles and the length of any side are known (AAS or ASA), or when the lengths of 2 sides and an angle opposite one of those 2 sides are known (SSA)

Law of Sines and Law of Cosines Law of Cosines: used to solve triangles when 2 sides and the included angle are known (SAS), or when all 3 sides are known (SSS)

Law of Sines and Law of Cosines Core Concepts (pages 274 and 275 in Student Journal) Area of a Triangle The area of any triangle can be found by ½ the product of the lengths of any 2 sides times the sine of their included angle.

Law of Sines and Law of Cosines Law of Sines Theorem sin A/a = sin B/b = sin C/c, where a, b and c are side lengths opposite angles A, B and C respectively

Law of Sines and Law of Cosines Law of Cosines Theorem If a, b and c are side lengths of a triangle opposite angles A, B and C respectively, then the following are true: a2 = b2 + c2 – 2bc(cos A) b2 = a2 + c2 – 2ac(cos B) c2 = a2 + b2 – 2ab(cos C)

Law of Sines and Law of Cosines Examples (page 276 in Student Journal) #4) Find the area of the triangle.

Law of Sines and Law of Cosines Solution #4) 61.8 units squared

Law of Sines and Law of Cosines #6) Solve the triangle.

Law of Sines and Law of Cosines Solution #6) m<C = 100 degrees, AB = 43.3, BC = 33.7