GEOMETRY MODULE 3 LESSON 2 PROPERTIES OF AREA OPENING EXERCISE Complete the Exploratory Challenge 1, parts a and b, in your workbook. DISCUSSION Consider Exploratory Challenge 2. Explain how we can calculate the area of the entire figure. The shapes on the ends are congruent and their areas are easily 1 found by × 푏푎푠푒 × ℎ푒푔ℎ푡. 2 The section in the middle is a rectangle. Its area is found by 푙푒푛푔푡ℎ × 푤푑푡ℎ. Once the separate areas are found, they are added together for the total area. Calculate the area. 1 퐴푟푒푎 표푓 푡푤표 푡푟푎푛푔푙푒푠 = 2 ( ) (3)(3) = 9 2 퐴푟푒푎 표푓 푡ℎ푒 푟푒푐푡푎푛푔푙푒 = (3)(7) = 21 푇표푡푎푙 퐴푟푒푎 = 9 + 21 = 30 Consider Exploratory Challenge 3. 퐴퐷 = 4, 퐷퐶 = 3, and 퐶퐹 = 2. Explain how we can calculate the area of the entire figure. Calculate the area of ∆퐴퐵퐶 and ∆퐷퐸퐹. Since ∆퐷퐺퐶 represents the overlapping area shared by ∆퐴퐵퐶 and ∆퐷퐸퐹, its area must be subtracted from the sum of the areas of ∆퐴퐵퐶 and ∆퐷퐸퐹. This avoids “counting twice.” MOD3 L2 1 Calculate the area of the figure ABGEF. 1 퐴푟푒푎 표푓∆퐴퐵퐶 = ( ) (7)(4) = 14 2 1 퐴푟푒푎 표푓∆퐷퐸퐹 = ( ) (5)(2) = 5 2 1 퐴푟푒푎 표푓∆퐷퐺퐶 = ( ) (3)(0.9) = 1.35 2 퐴푟푒푎 표푓퐴퐵퐺퐸퐹 = 14 + 5 − 1.35 = 17.65 Consider Exploratory Challenge 4. Explain how we can calculate the area of the entire figure. Subtract the area of the triangle from the area of the rectangle to find the area of the shaded region. Calculate the area. 퐴푟푒푎 표푓푅푒푐푡푎푛푔푙푒 = (12)(21.4) = 259.2 1 퐴푟푒푎 표푓 푇푟푎푛푔푙푒 = ( ) (9.6)(7.2) = 34.56 2 퐴푟푒푎 표푓푆ℎ푎푑푒푑 푅푒푔표푛 = 259.2 − 34.56 = 224.64 PROPERTITES OF AREA 1. The area of a set in the plane is a number, greater than or equal to zero, that measure the size of the set and not the shape. 2. The area of a rectangle is given by the formula 푙푒푛푔푡ℎ × 푤푑푡ℎ. The area of a triangle is given 1 by the formula × 푏푎푠푒 × ℎ푒푔ℎ푡. A polygonal region is the union of finitely many non- 2 overlapping triangular regions and has area the sum of the areas of the triangles. 3. Congruent regions have the same area. (Example: Challenge 1 and 2) 4. The area of the union of two regions is the sum of the areas minus the area of intersection. (Example: Challenge 3) 5. The area of the difference of two regions where one is contained in the other is the difference of the areas. (Example: Challenge 4) MOD3 L2 2 ON YOUR OWN Wooden pieces in the following shapes and sizes are nailed together to create a sign in the shape of an arrow. The pieces are nailed together so that the rectangular piece overlaps with the triangular piece by 4 inches. What is the area of the region in the shape of the arrow? 퐴푟푒푎 (퐴푟푟표푤) = 퐴푟푒푎(푅푒푐푡푎푛푔푙푒) + 퐴푟푒푎(푇푟푎푛푔푙푒) − 퐴푟푒푎 (푂푣푒푟푙푎푝) 퐴푟푒푎(푅푒푐푡푎푛푔푙푒) = (6)(24) = 144 Units must be converted to inches first! 1 퐴푟푒푎(푇푟푎푛푔푙푒) = ( ) (13)(16) = 104 2 퐴푟푒푎 (푂푣푒푟푙푎푝) = (6)(4) = 24 퐴푟푒푎 (퐴푟푟표푤) = 144 + 104 − 24 = 224 푛2 HOMEWORK Problem Set Module 3 Lesson 2, page 12 #1, #2, #3, and #4. Round answers to the nearest whole unit. Show all work in an organized and linear manner. DUE: Wednesday, February 13, 2017 MOD3 L2 3 .