Cosine Law, Similar Figures & Equivalent figures 70. Very important note: You must show all your work clearly and with great details.

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1 D Arcy McGee High School Major Assignment: Geometry Unit March &7, 017 Duration: 60 minutes Cosine Law, Similar Figures & Equivalent figures 70 Name: Solutions Section: MCU504- Very important note: You must show all your work clearly and with great details. 1. Ms. Peterson s farm land is represented by the diagram on the right. She decided to fence the front side (line BC) of her property. Determine how much Ms. Peterson will have to pay to build the fence. The fence costs $4.50/m. (10 marks) Using the Cosine Law: mbc ² = (10)(150)cos114 mbc = ( (10)(150)cos114 ) (use your calculator directly here) Very Important Note: You must use a calculator that displays the functions and numbers on the screen as you would write them on paper. Do not use a calculator that requires you to enter numbers before punching function keys. mbc = 7 m Cost of the fence is 7m x $4.50/m = $ Ms. Peterson will have to pay $ to fence the front side of her farm. CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page 1

2 . Sean and Ruth pitched a tent at a campground right in the heart of Kluane National Park. The tent is represented by the adjacent triangle ABD. In order to stabilize the tent, a rope (line AC) must be attached from the tip (A) of the tent to the ground (at point C). Determine the length of the rope AC. (10 marks) Considering the larger triangle ABC mbc = mbd + mdc mbc = 4.5 m +.5 m = 7m Using the Cosine Law: mac² = (. 4)(7)cos65 mac = ( (. 4)(7)cos65 ) (use you calculator right here) mac = 6.6 m The length of the rope AC is 6.6 m CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page

3 . A rectangle with 8 cm length and 6 cm width is equivalent to a triangle with a 1 cm base. Determine the height of the triangle? (5 marks) The 8cm x 6cm rectangle and the triangle with base 1 cm are equivalent They have the same area 1 height 8 x 6 = 48 = 6 x height (by dividing 1 by on the right hand side of the equation) height = 6 (by dividing by 6 on both sides) 8 = height The height of the triangle is 8 cm 4. A 8 cm by 6 cm by cm rectangular prism is equivalent to a square based pyramid. If the side length of the pyramid's base is 6 cm. Calculate the height of the pyramid. (5 marks) The 8cm x 6cm x cm rectangular prism and the square-based pyramid are equivalent They have the same volume area of base height 8 x 6 x = 6 6 height 144 = 6 height 144 = 144 = 1 x height (by dividing 6 by on the right hand side of the equation) height = 1 by dividing by 1 on both sides The height of the pyramid is 1 cm CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page

4 5. A cone and a cylinder are equivalent. The height of the cone is 4 cm and its volume is equal to 160 cm³. The radius of the cylinder is half the radius of the cone. Determine the total surface area of each solid. (Round your answer to the nearest unit) (10 marks). The cone and the cylinder are equivalent They have the same volume Volume (cone) = Volume (cylinder) = 160 cm³. Height of the cone = 4 cm area of base height Volume(cone) = πr² = (160) 4π = r² (160) 4π = r 8cm = r Radius of the cylinder (R) is half that of the cone R = 8 = 4 cm The height of the cylinder can be calculated with the volume formula: Area (base) x height = 160 π 4²x height = 160 height = 160 π 4² height of the cylinder = cm CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page 4

5 Total surface area of each solid Total Surface area of the cone: Total Surface Area of the cone = πrs + πr² Where s (slanted height of the cone) = radius² + height² Total Surface Area of the cone = π 8 8² + 4² + π 8² Total Surface Area of the cone = 87cm² Total Surface area of the cylinder: Total Surface Area of the cylinder = πrh + πr² Total Surface Area of the cylinder = π 4 + π 4² Total Surface Area of the cylinder = 905 cm² The total surface area of the cone is: 87cm² The total surface area of the cylinder is: 905 cm² CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page 5

6 6. The rectangle and the right trapezoid on the right are equivalent. What is the numerical value of the perimeter of the trapezoid? (10 marks) Two plane figures are equivalent they have the same area The cone and the cylinder are equivalent Area (rectangle) = Area (Trapezoid) (large base+small base) height Length x Width = (x+5+x ) 4 1(x ) = 1(x ) = (x+)4 1x 4 = (x+)4 (On the right hand side of the equation, divide 4 by ) 1x 4 = (x + ) 1x - 4 = 6x + 6 (by distributing to each term of (x + )) 1x 6x = x = 0 x = 5 CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page 6

7 The dimensions of the trapezoid are: Width = 4 Large base = x = 10 cm Small base = x (5) = 8 cm The slanted side (s) of the trapezoid can be calculated by using the Pythagorean formula: s² = 4² + (10 8)² s = 4² + ² s = 4.47 cm Therefore the perimeter of the trapezoid is = Perimeter of the trapezoid = 6.47 cm CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page 7

8 7. The rectangular prism, the cube and the sphere below are equivalent. What is the numerical value of the surface area of each solid? Round your answers to the nearest tenth of a unit. (10 marks) The three solids are equivalent They have the same volume Volume (rectangular prism) = Volume (cube) x = 8 x = 4 unit (4)(x+4)() = 4³ 8(x + 4) = 64 8 x + = 64 8 x = 64 8 x = Radius of the sphere: Volume (sphere) = Volume (cube) 4πRadius³ 4πRadius³ = 64 = 4³ ( 64) Radius³ = 4π (isolate Radius³then cross multiply by 64 and divide by4π ) Radius = (64) 4π =.5 unit The radius of the sphere is.5 unit CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page 8

9 Total surface area of each solid Total Surface area of the rectangular prism: Total surface area of the rectangular prism = (4 8) + ( 8) + ( 4) Total surface area of the rectangular prism = 11 unit² Total Surface area of the cube: Total surface area of the rectangular prism = 6(4 4) Total surface area of the cube = 96 unit² Total Surface area of the sphere: Total surface area of the sphere= 4πr² 4π(.5²) = 78.5 unit² Total surface area of the sphere = 78.5 unit² Conclusion: The total surface area of the rectangular prism is: 11 unit² The total surface area of the cube is: 96 unit² The total surface area of the sphere is: 78.5 unit² CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page 9

10 8. Figures A and B below are equivalent whereas figures B and C are similar. The area of figure C is 15 cm² greater than four times the area of figure A. Determine the perimeter of figure C. (10 marks) Two plane figures are equivalent they have the same area One plane figure is similar to another when the ratio of their areas is equal to a factor of k², whereas the ratio of their corresponding dimensions is equal to k The triangle A and the rectangle B are equivalent Area (triangle) = Area (rectangle) base height = Length x Width (x+) 10 = x.6 (x + )5 = 6x 5x + 10 = 6x 10 = 6x 5x 10 = x CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page 10

11 Finding the perimeter of figure C (larger rectangle) The larger rectangle (C) is similar to the smaller rectangle (B) the ratio of their areas is equal to k² factor. The ratio of their corresponding sides is equal to k. Area of figure A (triangle) = Area of figure B (rectangle) = 6 x 10 = 60 cm² (Since figure A and figure B are equivalent) Area of figure C (larger rectangle) = (60) = 75 cm² k²= Areaof larger rectangle Area of smaller rectangle = = 6.5 k = 6. 5 k =.5 Therefore, the dimensions of the larger triangle (figure C) are: Length =.5(10) = 5 cm Width =.5(6) = 15 cm Therefore, the perimeter of figure C is: (5 + 15) = 80 cm The perimeter of figure C is: 80 cm CST 11 - MATH MCU504: Major Assignment / Geometry Unit / Cosine Law, Similar & Equivalent Figures Page 11