We will determine1 how to use the geometric mean2 to find segment lengths in right triangles and apply similarity relationships in right triangles to solve.

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Presentation transcript:

We will determine1 how to use the geometric mean2 to find segment lengths in right triangles and apply similarity relationships in right triangles to solve problems. LEARNING OBJECTIVE Definition figure out special type of average where we multiply the numbers together and then take a square root. Declare the Objective A: Read the Objective to B. B: Define geometric mean to A CFU What are we going to learn today? What is “Geometric mean” mean? G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

ACTIVATE PRIOR KNOWLEDGE Remember the Concept Cross Products property. Solve for x use the Cross products property. 1. Solve for x, 2. Solve for x, = ? 4 x 8 3 x – = 8 x 6 15 Cross products property Cross products property 8 15 = x 6 4(x – 3) = x 8 Simplify. Divide each side by 6. 4x – 12 = 8x ACTIVATE PRIOR KNOWLEDGE 120 = 6x Subtract 4x from each side. 20 = x –12 = 4x Divide each side by 4. Make the Connection Students, you already know how to use cross product to solve for x. Today, we will learn how to use the geometric mean to find segment lengths . –3 = x

The geometric mean of two positive numbers is the positive square root of their product. To find the mean of a and b, consider the proportion . In this case, the means is: Geometric mean in right triangles: Find the geometric mean of the numbers 5 and 20 Check for Understanding B Explain to A: what is geometric mean and how do we find it? On your white board, find the geometric mean of: 3 and 75? How did you find it? x =15 B. x =16 x =14 B. x =17 CFU

𝒙 𝒃 = 𝒃 𝒄 Type 1: 𝒚 𝒂 = 𝒂 𝒄 Type 2: 𝒙 𝒉 = 𝒉 𝒚 Type 3: c How did I/you write the proportional ratios? How did I/you I solve for 𝒃 𝟐 , 𝒂 𝟐 , & 𝒉 𝟐 ? CFU 1 2 Identify the type of problem: Given as Numbers or Triangles Type. Steps for find the geometric mean 1 2 Write down proportional ratios. 3 Use cross product and find the square root. Three types of geometric means that can occur when you have a right triangle with an altitude drawn from the right angle. 𝒙 𝒃 = 𝒃 𝒄 Type 1: CONCEPT DEVELOPMENT 𝒚 𝒂 = 𝒂 𝒄 Type 2: c 𝒙 𝒉 = 𝒉 𝒚 Type 3:

𝒏 𝒃 = 𝒃 𝒄 𝒎 𝒉 = 𝒉 𝒏 𝒃 𝟐 =𝒏 * c 𝒉 𝟐 =𝒎 ∗𝒏 Check for Understanding How did I/you write the proportional ratios? How did I/you I solve for x? CFU 1 2 Identify the type of problem: Given as Numbers or Triangles Type. Write down proportional ratios. Use cross product and find the square root. Steps for find the geometric mean 1 2 3 1 Given m = 2 and n = 10. Find h. 2 Given m = 2 and n = 10. Find b. c a b n m h c a b n m h Check for Understanding A Explain to B: How did I write the ratio and find the h? CONCEPT DEVELOPMENT 𝒏 𝒃 = 𝒃 𝒄 𝒎 𝒉 = 𝒉 𝒏 𝒉 𝟐 =𝒎 ∗𝒏 𝒃 𝟐 =𝒏 * c c = 2 + 10 = 12

𝒎 𝒂 = 𝒂 𝒄 𝒂 𝟐 =𝒎 ∗𝒄 Given m = 27 and c = 30. Find a. How did I/you write the proportional ratios? How did I/you I solve for x? CFU 1 2 Identify the type of problem: Given as Numbers or Triangles Type. Write down proportional ratios. Use cross product and find the square root. Steps for find the geometric mean 1 2 3 1 Given m = 27 and c = 30. Find a. 2 Use the diagram to complete the equation. c a b n m h CONCEPT DEVELOPMENT Check for Understanding A Explain to B: How did I write the ratio and find the a? 𝒎 𝒂 = 𝒂 𝒄 𝒂 𝟐 =𝒎 ∗𝒄

Relevance Reason #1: Geometric Mean are used in finding the VIEWING ANGLE. A photographer wants to take a picture of a beach front. His camera has a viewing angle of 90° and he wants to make sure two palm trees located at points A and B in the figure are just inside the edges of the photograph. He walks out on a walkway that goes over the ocean to get the shot. If his camera has a viewing angle of 90°, at what distance down the walkway should he stop to take his photograph? Answer: 60 ft. RELEVANCE Relevance Reason #1: Know how to find geometric mean will help you do well on tests (PSAT, SAT, ACT, GRE, GMAT, LSAT, etc..). Sample Item Check for Understanding Does anyone else have another reason why it is relevant to use verb tense correctly? Which reason is most relevant to you? Why? Find the geometric mean of 4 and 36. If necessary, give the answer in simplest radical form.? 11 = x 10 = x 12 = x 13 = x

What did you learn today about how to use the geometric mean to find segment lengths in right triangles? Word Bank Geometric Mean Right Triangle Segment Lengths SUMMARY CLOSURE Today, I learned how to __________________ ______________________________________________________________.