Right Triangles and Trigonometry Chapter 8

8.1 Geometric Mean Geometric mean: Ex: Find the geometric mean between 5 and 45 Ex: Find the geometric mean between 8 and 10

If an altitude is drawn from the right angle of a right triangle If an altitude is drawn from the right angle of a right triangle. The two new triangles and the original triangle are all similar. B A C D

The altitude from a right angle of a right triangle is the geometric mean of the two hypotenuse segments B Ex: A C D

The leg of the triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent B Ex: A C D

Find the geometric mean between

Find c, d, and e.

Find e and f . (round to the nearest tenth if necessary)

8.2 Pythagorean Theorem and its Converse When and why do you use the Pythagorean Theorem? When: given a right triangle and the length of any two sides Why: to find the length of one side of a right triangle When do you use the Pythagorean Theorem Converse? When: you want to determine if a set of sides will make a right triangle

Pythagorean Theorem: When c is unknown: When a or b is unknown: c a2 + b2 < c2 obtuse a2 + b2 > c2 acute a b When c is unknown: When a or b is unknown: x 5 14 7 3 x

Converse: the sum of the squares of 2 sides of a triangle equal the square of the longest side 8, 15, 16 Pythagorean Triple: 3 lengths with measures that are all whole numbers & that always make a right triangle 3, 4, 5 5, 12, 13 7, 24, 25 9, 40, 41 Not =, so not a right triangle

A. Find x.

B. Find x.

A. Determine whether 9, 12, and 15 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.

B. Determine whether 10, 11, and 13 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.

8.3 Special Right Triangles 30-60-90 Short leg is across from the 30 degree angle Long leg is across from the 60 degree angle Ex: 14 x 30 y

Find x and y.

Find x and y.

45-45-90 The legs are congruent Ex: Ex: 6 x x x 8

A. B.

Find x and y.

The length of the diagonal of a square is centimeters The length of the diagonal of a square is centimeters. Find the perimeter of the square.

8.4 Trigonometry In Right triangles

A. Express sin L, cos L, and tan L as a fraction and as a decimal to the nearest ten thousandth.

Find the value to the ten thousandth. Sin 15 Tan 67 Cos 89.6

Find the measure of each angle to the nearest tenth of a degree Cos T = .3482 Tan R = .5555 Sin P = .6103

Find y.

Find the height of the triangle.

When you need to find the angle measure- set up the problem like normal Then hit the 2nd button next hit sin, cos or tan (which ever you are using) then type in the fraction as a division problem, hit = Find angle P.

Find angle D.

8.5 Angles of Elevation and Depression Draw a picture and solve using trigonometry. Mandy is at the top of the Mighty Screamer roller coaster. Her friend Bryn is at the bottom of the coaster waiting for the next ride. If the angle of depression from Mandy to Bryn is 26 degrees and The roller coaster is 75 ft high, what is the distance from Mandy to Bryn?

Mitchell is at the top of the Bridger Peak ski run Mitchell is at the top of the Bridger Peak ski run. His brother Scott is looking up from the ski lodge. If the angle of elevation from Scott to Mitchell is 13 degrees and the ground distance from Scott to Mitchell is 2000 ft, What is the length of the ski run?

An observer located 3 km from a rocket launch site sees a rocket at an angle of 38 degrees. How high is the rocket at that moment?

A kite is flying at an angle of elevation of 40 degrees A kite is flying at an angle of elevation of 40 degrees. All 50 m of string have been let out. What is the height of the kite?

Two buildings on opposite sides of the street are 40 m apart Two buildings on opposite sides of the street are 40 m apart. From the top of the taller building, which is 185 m tall, the angle of depression to the top of the shorter building is 13 degrees. How high is the shorter building?