By: Andrew Shatz & Michael Baker Chapter 15. Chapter 15 section 1 Key Terms: Skew Lines, Oblique Two...
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By: Andrew Shatz & Michael Baker Chapter 15
By: Andrew Shatz & Michael Baker Chapter 15. Chapter 15 section 1 Key Terms: Skew Lines, Oblique Two lines are skew iff they are not parallel and do not
Chapter 15 section 1 Key Terms: Skew Lines, Oblique Two lines
are skew iff they are not parallel and do not intersect. (lines ST
and UV are skew) A line and a plane, or two planes that are neither
parallel nor perpendicular are said to be Oblique.
Slide 3
If two points lie in a plane, the line that contains them lies
in the plane.
Slide 4
If two planes intersect, they intersect a line.
Slide 5
Definitions Two planes, or a line and a plane, are parallel iff
they do not intersect.
Slide 6
Definitions (continued) A line and a plane are perpendicular
iff they intersect and the line is perpendicular to every line in
the plane that passes through the point of intersection. Line AB is
perpendicular to lines: BE, BD, and BC
Slide 7
Definitions (continued) Two planes are perpendicular iff one
plane contains a line that is perpendicular to the other
plane.
Definitions Polyhedron: a solid bounded by parts of
intersecting planes Rectangular Solid: a polyhedron that has six
rectangular faces Cube: a rectangular solid with equal
dimensions
Slide 10
Slide 11
Vertex A and Vertex G are an example of opposite vertices
Slide 12
l: Length w: Width h: Height d: Diagonal
Slide 13
Chapter 15 section 3 Key Terms: Bases, Prism, Lateral Faces,
Lateral Edges, Right Prism, Oblique Prism, Net, Lateral Area, Total
Area
Slide 14
Prism: A solid geometric figure whose two end faces are
similar, equal, and parallel figures, and whose sides are
parallelograms.
Slide 15
Lateral face Lateral edge
Slide 16
Base The figure above is a net of a triangular prism. The
lateral area of a prism is the sum of the areas of its lateral
faces. The total area of a prism is the sum of its lateral area and
the areas of its bases Lateral Face
Slide 17
If the lateral edges of a prism are oblique to the planes of
its bases the prism is an oblique prism. Right Prism Oblique
Prism
An altitude of a prism is a line segment that connects the
planes of its bases and that is perpendicular to both of them.
Slide 20
The volume of an object is the amount of space that it
occupies. A cross section of a geometric solid is the intersection
of a plane and the solid.
Slide 21
Cavalieris Principle Consider two geometric solids and a plane.
If every plane parallel to this plane that intersects one of the
solids also intersects the other so that the resulting cross
sections have equal areas, then the two solids have equal
volumes.
Slide 22
Extra Lesson Finding values of Prisms and Pyramid using limited
information.
Pyramids A pyramid is a polyhedron in which the base is a
polygon, and the sides are triangles leading up to the apex. The
pyramid lies on its base. All other faces are lateral faces, and
the edges they intersect at are lateral edges. All of the lateral
edges meet at a single point called the apex.
Cones A cone is a figure that has a circular base, and all
segments lead up to its apex. The line segment connecting the apex
to the center of the of the base is called the axis. If the axis is
perpendicular to its base, then the cone is a right cone. If it is
oblique to its base, then it is an oblique cone.
Slide 30
Types of Cones Right ConeOblique Cone
Slide 31
Cylinders A cylinder is a figure that has two congruent and
parallel circular bases that are connected. A cylinder has two flat
bases, and a curved side which is its lateral surface. The axis
connects the centers of the bases. A cylinder be right or oblique
depending on its axes with respect to its bases.
Slide 32
Chapter 15 section 7 Key Terms: Sphere, Center, Radius,
Diameter
Slide 33
Spheres A sphere is a set of all points in space that are at a
given distance from a given point. The center, radius, and diameter
have the same meanings for spheres as they do for circles.
Similar Solids Two geometric solids are similar if they have
the same shape. Theorem 85: The ratio of the surface areas of two
similar rectangular solids is equal to the square of the ratio of
any pair of corresponding dimensions. Theorem 86: The ratio of the
volumes of two similar rectangular solids is equal to the cube of
the ratio of any pair of corresponding dimensions.
The Regular Polyhedra A regular polyhedron is a convex solid
having faces that are congruent regular polygons, and having an
equal number of polygons that meet at each vertex. Examples:
Equilateral Triangles Tetrahedron, 4 FacesOctahedron, 8 Faces
Icosahedron, 20 Faces
Insight! A huge part of this chapter is remembering the various
formulas. Without knowing them, it will be nearly impossible to
complete problems. Remembering the regular polyhedra is also a
large part. The main thing is knowing how many sides there are, and
what shapes the faces are. Lastly, knowing the vocabulary of the
chapter will make it immensely easier to solve problems.