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Chapter 15

Chapter 15. By : Alison Johnson & Caroline Foster. 15.1 Lines and Planes in Space. If two points lie in a plane, the line that contains them lies in the plane. If two planes intersect, they intersect in a line. Two lines are skew iff they are not parallel and do not intersect.

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Chapter 15

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  1. Chapter 15 By : Alison Johnson & Caroline Foster

  2. 15.1 Lines and Planes in Space • If two points lie in a plane, the line that contains them lies in the plane. • If two planes intersect, they intersect in a line. • Two lines are skewiff they are not parallel and do not intersect. • Two planes, or a line and a plane, are parallel iff they do not intersect. Skew Lines Two planes intersecting at a line

  3. 15.1 Lines and Planes in Space • 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. • Two planes are perpendicular iff one plane contains a line that is perpendicular to the other plane. • A line and a plane (or two planes) that are nether parallel nor perpendicular are said to be oblique.

  4. Extra Theorems • *A line and a point not on a line determine a plane • Two intersecting lines determine a plane • Two parallel lines determine a plane • If three lines are perpendicular to a line at the same point, the three lines are coplanar • Planes perpendicular to the same line are parallel to one another • Lines perpendicular to the same plane are parallel to one another • A plane perpendicular to one of two parallel lines is perpendicular to both of them

  5. 15.2 Rectangular Solids -A polyhedron is a solid bounded by parts of intersecting planes. -Intersecting planes form the polygonal regions that are the faces. -Their sides are the edges of the polyhedron and its vertices are it vertices. -A rectangular solid is a polyhedron that has six rectangular faces. -Two vertices of the solid that are not vertices of the same face are opposite vertices.

  6. 15.2 Rectangular Solids -A line segment that connects two opposite vertices of a rectangular solid is a diagonal of the solid. -The lengths of the three edges of a rectangular solid that meet at one of its vertices are the dimensions of the solid and are usually called itslength, width, and height. Diagonal

  7. 15.2 Rectangular Solids • The length of a diagonal of a rectangular solid with dimensions x, y, and z is: • The length of the diagonal of a cube with edge lengths of e is e3.

  8. 15.3 Prisms -Every prism has two congruent faces, its bases, which lie in parallel planes. -The rest of the faces in a prism are its lateral faces. -The edges in which the lateral faces intersect one another are its lateral edges. -If the lateral edges of a prism are perpendicular to the planes of its bases, then it is a right prism. -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

  9. 15.3 Prisms -The lateral areaof a prism is the sum of the areas of its lateral faces. -The total areaof a prism is the sum of its lateral area and the areas of its bases. Nets: Area Formula: Ph+2B

  10. 15.4 Volume of a 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. -The volume of an object is the amount of space that it occupies. -A cross sectionof a geometric solid is the intersection of a plane and a solid. Cavalieri’s 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.

  11. 15.4 Volume of a Prism -The volume of any prism is the product of the area of its base and its altitude. V=Bh -The volume of a rectangular solid is the product of its length, width, and height. V=lwh -The volume of a cube is the cube of its edge. V=e³

  12. 15.5 Pyramids • Base- face of pyramid that lies on a plane • Lateral faces- rest of faces • Lateral edges- the edges at which the lateral faces intersect • Apex- the point at which the lateral edges meet • Altitude- the perpendicular line segment connecting the apex to the plane of its base

  13. 15.5 Pyramids • Surface Area: ½Pl+B • The volume of any pyramid is one-third the product of the area of its base and its altitude Volume: ⅓Bh

  14. Helpful Formulas to Know Lateral height Lateral edge Lateral edge Lateral height h h r a ½ side These can be very helpful when finding the areas of pyramids and prisms s s Area equilateral triangle s² s

  15. Other Helpful Formulas 30- 60 Right Triangle Isosceles Right Triangle 2s s s s s s

  16. 15.6 Cones • Lateral surface- curved surface • Circular region- union of a circle and its interior • Apex • altitude • Base- face of cone that lies on plane (circle) • Axis- the line segment that connects the apex of a cone to the center of its base • A cone is right or oblique depending on whether or not the axis is perpendicular to the base

  17. 15.6 Cones • Surface area: • The volume of a cone is one-third the product of the area of its base and its altitude Volume: ⅓

  18. 15.6 Cylinders • Have 2 bases • Lateral surface- curved surface • Altitude • Axis- line segment connecting the centers of its bases • Right Cylinder • Oblique Cylinder

  19. 15.6 Cylinders • Surface Area: 2 • The volume of a cylinder is the product of the area of its base and its altitude Volume: h

  20. 15.7 Spheres • Sphere- set of all points in space that are at a given distance from a given point • Has a center, diameter, and radius

  21. 15.7 Spheres • Surface Area: 4 • Volume: 4/3³

  22. 15.8 Similar Solids • 2 geometric solids are similar if they have the same shape • If the ratio of corresponding dimensions of 2 similar solids is r, then the ratio of their surface areas is r² • If the ratio of the corresponding dimensions of 2 similar solids is r, then the ratio of their volumes is r³

  23. 15.9 Regular Polyhedra • Regular polyhedron- convex solid having faces that are congruent regular polygons and having an equal number of polygons that meet at each vertex

  24. 15.9 Regular Polyhedra Tetrahedron-equilateral Triangles with 4 faces Octahedron-equilateral Triangles with 8 faces Icosahedron-equilateral Triangles with 20 faces Cube-squares with 6 faces Dodecahedron-regular Pentagons with 12 faces

  25. Formulas For Geometric Solids

  26. Helpful Formula • V+F-E=2 • This equation helps when trying to determine the number of one of these parts • This equation works for all Prisms, Pyramids, and Polyhedrons V= 8 F= 6 E=12 8+6-12=2

  27. Chapter Summary • In this chapter we learned all about points, lines, and planes when they are found in three dimensions. We discovered the surface area and volume formulas for many different Geometric Solids • We also discovered that for two geometric solids to be similar they have a similar shape, but in different proportions. In addition, similar solids have proportional surface areas and volumes too. • We also discovered the 5 regular Polyhedrons and the fact that they all have congruent faces and the same number of polygons that meet at each vertex

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