GEOMETRY MODULE 3 LESSON 7 GENERAL PYRAMIDS AND CONES AND THEIR CROSS-SECTIONS
OPENING EXERCISE Complete the opening exercise on page 49 in your workbook.
General Cylinders: 1 and 6 Prisms: 2 and 7 Figures that come to a point with a polygonal base: 4 and 5 Figures that come to a point with a curved base: 3 and 8
DISCUSSION Rectangular Pyramid: Given a rectangular region B in a plane E and a point V not in E, the rectangular pyramid with base B and vertex V is the collection of all segments VP for any point P in B.
Image 4 from the Opening Exercise is an example of a rectangular pyramid.
MOD3 L7 1 General Cone: Let B be a region in a plane E and V be a point not in E. The cone with base B and vertex V is the union of all 푉푃̅̅̅̅ for all point P in B.
Images 2, 4, 5, and 8 from the Opening Exercise are examples of general cones.
The definition for rectangular pyramid and general cone are essentially the same. What is the only difference? The only difference is that a rectangular pyramid has a rectangular base. A general cone can have any region for a base.
Important Notes about Pyramids and Cones A general cone is named by its base. o A general cone with a disk as a base is called a circular cone. o A general cone with a polygonal base is called a pyramid. Examples of this include a rectangular pyramid or a triangular pyramid. A general cone whose vertex lies on the perpendicular line to the base and that pass through the center of the base is a right cone (or a right pyramid if the base is polygonal). Video: How to Create a Cone from a Right Triangle https://youtu.be/Tz7TQljoiM8
Consider the pyramid to the right. Name all the lateral faces. AVB, BVC, CVD, DVA Name all the lateral edges. VA, VB, VC, VD Name the base. ABCD
MOD3 L7 2 3D DILATIONS We studied dilations in two dimensions, but it turns out that dilations behave similarly in three-dimensional space.
A dilation of three-dimensional space with center O and scale factor r is defined the same way it is in two-dimensions. The dilation maps O to itself and maps any other point X to the point 푋’ on ray OX so that 푶푿′ = 풓푶푿 Note: Each cross-section of a general cone is similar to its base. The angles are congruent and the side lengths are proportional. Each cross-section is also parallel to its base.
PRACTICE Consider Example 1 on page 51 of your workbook. If ∆퐴′퐵′퐶′ is a cross-section of the pyramid and thus parallel to the base, what can be said about 퐴퐵̅̅̅̅ and 퐴̅̅̅′̅퐵̅̅′?
푉퐴′ 3 Find the scale factor for . 푉퐴 5 Apply the Scaling Principle for Area.
3 2 퐴푟푒푎 ∆퐴′퐵′퐶′ = ( ) 퐴푟푒푎 ∆퐴퐵퐶 5
3 2 퐴푟푒푎 ∆퐴′퐵′퐶′ = ( ) (25) 5 퐴푟푒푎 ∆퐴′퐵′퐶′ = 9푚푚2
Consider Example 2 in your workbook. 푉푋′ 12 2 Find the scale factor for . = 푉푋 18 3 Apply the Scaling Principle for Area.
2 2 퐴푟푒푎 ∆퐴′퐵′퐶′ = ( ) 퐴푟푒푎 ∆퐴퐵퐶 3
2 2 28 = ( ) 퐴푟푒푎 ∆퐴퐵퐶 3 9 퐴푟푒푎 ∆퐴퐵퐶 = 28 ( ) = 63푚푚2 4
MOD3 L7 3 ON YOUR OWN Attempt Exercise 1 on page 52. The area of the base of a cone is 16 and the height is 10. Find the area of a cross-section that is distance 5 from the vertex.
5 1 Scale Factor: = 10 2
1 2 퐴푟푒푎 표푓 퐶푟표푠푠 − 푆푒푐푡𝑖표푛 = ( ) 퐴푟푒푎 퐵푎푠푒 2 1 퐴푟푒푎 표푓 퐶푟표푠푠 − 푆푒푐푡𝑖표푛 = (16) 4 퐴푟푒푎 표푓 퐶푟표푠푠 − 푆푒푐푡𝑖표푛 = 4푢푛𝑖푡푠2
DISCUSSION General Cone Cross-Section Theorem: If two general cones have the same base area and the same height, then the cross-sections for the general cones the same distance from the vertex have the same area.
PROOF Let the bases of cones B and C be such that 퐴푟푒푎(퐵) = 퐴푟푒푎(퐶), the height of each cone be h and the distance from each vertex to 퐵’ and to 퐶’ are both ℎ’. Write an expression for 퐴푟푒푎(퐵′) using the Scaling Principle for Area. ℎ′ 2 퐴푟푒푎(퐵′) = ( ) 퐴푟푒푎(퐵) ℎ Write an expression for 퐴푟푒푎(퐶′) using the Scaling Principle for Area. ℎ′ 2 퐴푟푒푎(퐶′) = ( ) 퐴푟푒푎(퐶) ℎ Since 퐴푟푒푎(퐵) = 퐴푟푒푎(퐶), make a substitution to show 퐴푟푒푎(퐵′) = 퐴푟푒푎(퐶′),
ℎ′ 2 퐴푟푒푎(퐵′) = ( ) 퐴푟푒푎(퐶) = 퐴푟푒푎 (퐶′) ℎ 퐴푟푒푎(퐵′) = 퐴푟푒푎 (퐶′)
MOD3 L7 4 PRACTICE Consider Exercise 2 on page 53.
퐵′퐶′ 2 3 Find the scale factor for . √ 퐵퐶 3√2
Apply the Scaling Principle for Area.
퐴푟푒푎(퐴′퐵′퐶′퐷′) = 푟2 퐴푟푒푎 (퐴퐵퐶퐷) 2 2 3 퐴푟푒푎(퐴′퐵′퐶′퐷′) = ( √ ) 퐴푟푒푎 (푇푈푉푊푋푌푍) 3√2 2 2√3 12 퐴푟푒푎(퐴′퐵′퐶′퐷′) = ( ) (30) = ( ) (30) 3√2 18 퐴푟푒푎(퐴′퐵′퐶′퐷′) = 20푢푛𝑖푡푠2
HOMEWORK Problem Set Module 3 Lesson 7, page 54
1 #1, #2 Area of a Trapezoid = (푏 + 푏 )ℎ, #3, #6, #7, and #9 2 1 2 DUE: Wednesday, February 22, 2017
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