Geometry Chapter 9 Right Triangles and Trigonometry Lesson

Geometry Chapter #9 Right Triangles and Trigonometry

Lesson #9. 1 Similar Right Triangles

Right Triangles All right triangles have one right angle The longest side of a right triangle is the hypotenuse. It is always opposite the right angle. The sides that form the right angle are called legs.

Altitude of a Triangle An altitude of a triangle is the perpendicular segment from a vertex to the opposite side (or to a line that contains the opposite side). Every triangle has 3 altitudes

The three altitudes of a triangle are concurrent. The point of concurrency of the altitudes is called the orthocenter of the triangle. The orthocenter may lie inside, on, or outside the triangle. In an acute triangle, the In an obtuse triangle, the In a right triangle, the orthocenter is ON the triangle. INSIDE the triangle. OUTSIDE the triangle.

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.

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.

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 the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

Lesson #9. 2 The Pythagorean Theorem

Theorem #9. 4 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.

25 9 3 5 4 16 25 = 9 + 16 25 = 25

Pythagorean Triples are sets of whole numbers, a, b, and c, that work in the Pythagorean Theorem. Here are some that you should memorize: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 ANY multiples of these Triples are also Triples. (ex: 6, 8, 10 ---multiple 3, 4, 5 each by 2)

Lesson #9. 3 The Converse of the Pythagorean Theorem

Theorem #9. 5 If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, sides then the triangle is a right triangle. This theorem can be used to prove or verify that a triangle is a right triangle.

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, sides then the triangle is acute The triangle is acute.

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, sides then the triangle is obtuse The triangle is obtuse.

Lesson #9. 4 Special Right Triangles

Special Right Triangle Theorems Theorem #9. 8

Special Right Triangle Theorems Theorem #9. 9

Lesson #9. 5 Trigonometric Ratios

6 Trigonometric Functions

A mnemonic device that you can use to memorize the definition of the first three trig functions is SOH CAH TOA

Another mnemonic device that you can use to memorize the definition of the first three trig functions is

First, how could you find the length of hypotenuse AC? Use the Pythagorean Theorem:

hypotenuse opposite adjacent Next, how can you evaluate the 6 trig functions of angle A? Use the trig ratios from angle A’s point of view:

Lesson #9. 6 Solving Right Triangles

* Example: Solve the right triangle to find x, h, and missing angle A. Since this is not a special right triangle, I must use my calculator to find the values of the trig functions of the angles. A opp C IMPORTANT: My calculator must be in degrees (not radians). Round trig values correctly to 4 decimal places, and round final side lengths or angles to 1 decimal place. hyp h adj B Since two angles are 28 and 90, the third angle (angle A) must be 62 because the 3 angles must add up to 180 and 180 -90 -28 = 62. h = 25. 6 x = 22. 6

* Solving a right triangle means finding ALL missing side lengths AND angle measures. Example: Solve the right triangle to find x, y, and missing angle A. hyp Now notice that this is a 30 -60 -90 right triangle. I can use the special right triangles short-cut to solve this problem. opp adj Since two angles are 60 and 90, the third angle (angle A) must be 30 because the 3 angles must add up to 180 and 180 -60 -90 = 30. 2 y = 10 2 2 2 y=5 2

Example in real life: Wind speed affects the angle at which a kite flies. You are flying a kite 4 feet off the ground using 500 feet of line. At a wind speed of 35 miles per hour, the kite will make an angle with a line parallel to the ground of 48 degrees. How high off the ground is the kite. Add on the 4 feet for the height at which you are holding it, and the kite is about 376 feet off the ground.