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Slicing a Right Rectangular Pyramid with a Plane





 


Videos and solutions to help Grade 7 students learn how to describe rectangular regions that result from slicing a right rectangular prism by a plane perpendicular to one of the faces.

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Common Core For Grade 7

New York State Common Core Math Grade 7, Module 6, Lesson 17


Lesson 17 Student Outcomes


• Students describe polygonal regions that result from slicing a right rectangular pyramid by a plane perpendicular to the base and by another plane parallel to the base.

Lesson 17 Summary


• The vertex of a right rectangular pyramid lies on the line perpendicular to the base at its center (the intersection of the rectangle base’s diagonals); a pyramid that is not a right rectangular pyramid will have a vertex that is not on the line perpendicular to the base at its center.
• Students should visualize slices made perpendicular to the base of a pyramid by imagining a piece of paper passing through a given segment on a lateral face perpendicularly towards the base. Consider the outline the slice would make on the faces of the pyramid.
• Slices made parallel to the base of a right rectangular pyramid are scale drawings (i.e., reductions) of the rectangular base of the pyramid.

Lesson 17 Classwork

Opening
Rectangular Pyramid: Given a rectangular region B in a plane A, 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. It can be shown that the planar region defined by a side of the base B and the vertex V is a triangular region, called a lateral face.

Example 1
Use the models you built to assist in a sketch of a pyramid: Though you are sketching from a model that is opaque, use dotted lines to represent the edges that cannot be seen from your perspective.

Example 2
Sketch a right rectangular pyramid from three vantage points: (1) from directly over the vertex, (2) facing straight on to a lateral face, and (3) from the bottom of the pyramid. Explain how each drawing shows each view of the pyramid.

Example 3
Assume the following figure is a top-down view of a rectangular pyramid. Make a reasonable sketch of any two adjacent lateral faces. What measurements must be the same between the two lateral faces? Mark the equal measurements. Justify your reasoning for your choice of equal measurements.



Example 4
a. A slicing plane passes through segment a parallel to base B of the right rectangular pyramid below. Sketch what the slice will look like into the figure. Then sketch the resulting slice as a two-dimensional figure. Students may choose how to represent the slice (e.g., drawing a 2D or 3D sketch or describing the slice in words).
b. What shape does the slice make? What is the relationship between the slice and the rectangular base of the pyramid?

Example 5
A slice is to be made along segment perpendicular to base B of the right rectangular pyramid below. a. Which of the following figures shows the correct slice? Justify why each of the following figures is or is not a correct diagram of the slice.
b. A slice is taken through the vertex of the pyramid perpendicular to the base. Sketch what the slice will look like into the figure. Then, sketch the resulting slice itself as a two-dimensional figure.


 

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