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GEOMETRY MODULE 2 LESSON 31 USING TRIGONOMETRY TO DETERMINE AREA OPENING EXERCISE Three triangles are presented below. Determine the areas for each triangle, if possible. If it is not possible, describe what is need in order to determine the area. 1 (5)(12) = 30 2 1 π΄πππ βπ·πΈπΉ = (20)(8) = 80 2 π΄πππ βπ΄π΅πΆ = There is not enough information to find the height of βπΊπ»πΌ and thus we cannot find its area. ο· Is there a way to find the missing information? No DISCUSSION Consider βπΊπ»πΌ given a value for side GH. Could we find the area of βπΊπ»πΌnow? MOD2 L31 1 Find a value for x. β2 + π₯ 2 = 72 β2 + (15 β π₯)2 = 122 β2 = 49 β π₯ 2 β2 = 144 β (15 β π₯)2 49 β π₯ 2 = 144 β (15 β π₯)2 49 β π₯ 2 = 144 β (225 β 30π₯ + π₯ 2 ) 49 β π₯ 2 = 144 β 225 + 30π₯ β π₯ 2 130 = 30π₯ π₯= 13 3 Substitute to find height. 13 2 β = 49 β ( ) 3 169 β2 = 49 β 9 272 β2 = 9 2 β= β 272 4β17 = 9 3 Finally, we calculate area. 1 4β17 π΄πππ = (15) ( ) = 10β17 2 3 Letβs consider representing area for such triangles in general. π΄πππ = 1 πβ 2 1 π π΄πππ = 2 π β (π) Multiplication by 1 1 β π π΄πππ = 2 π (1) (π) 1 β π΄πππ = 2 π π (π) Terms are rearranged by the value is the same. MOD2 L31 2 ο· β With respect to π, what does π represent? β = sin π π ο· Make the substitution to the last line from above. π΄πππ = 1 ππ sin π 2 where a is the base and b is the βotherβ side to π ο· Which part of the expression above represents the height? β = π sin π PRACTICE A farmer is planning how to divide his land for planting next yearβs crops. A triangular plot of land is left with two known side lengths measuring 500m and 1700 m. These sides create a 30° angle. Draw and label the triangle then find the area. 1 π΄πππ = ππ sin π 2 1 π΄πππ = (1700)(500) sin 30 2 π΄πππ = 212,500 π π. πππ‘πππ WORKBOOK Exercise 1: A real estate developer and her surveyor are searching for their next piece of land to build on. They each examine a plot of land in the shape of βπ΄π΅πΆ. The real estate developer measures the length of AB and AC and finds them to both be approximately 4,000 feet, and the included angle has a measure of approximately 50°. The surveyor measures the length of AC and BC and finds the lengths to be approximately 4,000 feet and 3,400 feet, respectively, and measures the angle between the two sides to be approximately 65°. ο· Draw a diagram that models the situation. Label all lengths and angle measures. MOD2 L31 3 ο· Find the area of the triangle. Round your answer to the nearest whole number. NOTE: There are two possible answers. 1 ππ sin π 2 1 π΄πππ = (4000)(4000) sin 50 2 π΄πππ = π΄πππ = 6,128,356 π π. ππππ‘ 1 ππ sin π 2 1 π΄πππ = (3400)(4000) sin 65 2 π΄πππ = π΄πππ = 6162893 π π. ππππ‘ SUMMARY ο· To determine the area of a triangle when height is not provided use π π¨πππ = π ππ π¬π’π§ π½ , where a is the base and b is the βotherβ side to π. NOTE: The angle used must be the angle created by the two known sides. HOMEWORK Problem Set Module 2 Lesson 31, page 234 #1 thru #4, #6, #8, and #10 Show all work in an organized and linear manner. DUE: Tuesday, Jan 30, 2017 MOD2 L31 4