The arithmetic mean is relevant any time several quantities add together to produce a total. The arithmetic mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same total?“ In the same way, the geometric mean is relevant any time several quantities multiply together to produce a product. The geometric mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same product?" For example, suppose you have an investment which earns 10% the first year, 50% the second year, and 30% the third year. What is its average rate of return? The answer is the geometric mean (1.10 x 1.60 x 1.20)^(1/3). If you calculate this geometric mean you get approximately 1.283, so the average rate of return is about 28% Any time you have a number of factors contributing to a product, and you want to find the "average" factor, the answer is the geometric mean. The example of interest rates is probably the application most used in everyday life.

Chapter 7:Right Triangles 7.3: Use Similar Right Triangles and Geometric Mean

Geometric Mean Use notecards . . . Name the similar triangles: ΔABC ~ ΔADB ~ ΔBDC Use notecards . . .

Geometric Mean In the proportion = , a and d are called the extremes and b and c are called the means. When the means are the same, = , then x is called the geometric mean between a and d. Rem! The arithmetic mean or average is different from the geometric mean

Geometric Mean When you are asked to find the geometric mean between two numbers, use the proportion = , and fill in the “a” and “d” values with the two numbers to solve for “x”. Find the Geometric Mean. a. 4 and 9 b. and c. 5 and 10 x = 6

Geometric Mean x y x z y z YOU HAVE TO KNOW THIS!!!!!!! Complete each statement. h is the geometric mean between ______ and _____. (We call this one the “heartbeat”) 2. a is the geometric mean between _______and _______. 3. b is the geometric mean between _______and ______. x y x z y z YOU HAVE TO KNOW THIS!!!!!!! 3 Geometric Means: Leg 1 Leg 2 Heartbeat: altitude Seg. 1 Seg. 2 hypotenuse Look at the triangles and the angles involved

Geometric Mean Find each variable: Hint: Label L1, L2, etc. 5. 6. 5. 6. 7.5 6 + x 18 13.5 x = x = 7.5 y = y = z = 10.062 z =

Do odds on ws 5. If JK = 16 and OK = , find KM. 6. If OK = 6 find OM, if KM = 10. 7. If JK = 12 and KM = 16 find OK. Do odds on ws

Do odds On WS

Geometric Mean To find the height of her school building, Mieko held a book near her eye so that the top and bottom of the building were in line with the edges of the cover. If Mieko’s eye level is 5 feet off the ground and she is standing about 10 feet from the building, how tall is the building? We are finding the height of the building, so we need to find the length of BD and will add it to AD. Let’s say BD = x… x = 20 …but the question asks for how tall the building is so we add: AD + BD = height of the building 5 + 20 = 25 foot tall building

Remember Chief Sohcahtoa Trigonometric Ratio Abbreviation Definition Sine of A Sin A opp. A = a Hypotenuse c Cosine of A Cos A adj. to A = b Hypotenuse c Tangent of A Tan A opp. A = a adj. to A b

Examples Using Sohcahtoa: B 13 12 5 1. sin A = sin B = cos A = cos B = tan A = tan B =

Examples: Use a calculator. Find the following, rounding to 4 decimal places. sin 27 = B) tan 32 = C) cos 72 = D) sin 48 = .4540 .6249 .7431 .3090 mode!

Examples: inverse function A) sin B = B) cos E = C) tan I = Find the measure of the acute angles given the same trigonometric ratio. A) sin B = B) cos E = C) tan I = H I G E F D C B A 62° X = 67.38 ~ 67° 36.89 ~ 37° inverse function How can I find angles with this???? 2nd sin 15/17

Tan 50 = x / 4.8

Practice In class 7.5-7.6 Trig ratios HW Geometric mean and Trig application problems