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4. Area(^ABD) Area(^BCD) 5. Area of ABCD
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CHAPTER 16 Proofs of Important Theorems
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The area of a regular polygon is equal to one-half the product of its perimeter and apothem.
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Given: Regular polygon ABCDE . . . having center O, apothem of length r, perimeter p 1 To Prove: Area of ABCDE . . . 2rp Plan: By joining each vertex to the center, congruent triangles are obtained, the sum of whose areas equals the area of the regular polygon.
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Statements 1. Draw OA, OB, OC, OD , OE, . . . . 2. r is the altitude of each triangle formed. 3. Area of ^AOB ^BOC
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Reasons 1. A straight line segment may be drawn between two points. 2. Apothems of a regular polygon are congruent. 3. The area of a triangle equals one-half the product of the length of its base and altitude.
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^COD ................ 4. Area of regular polygon ABCDE . . .
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4. If equals are added to equals, the sums are equal.
... . . .) 5. The whole equals the sum of its parts.
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6. Area of ABCDE . . .
6. Substitution Postulate
Extending Plane Geometry into Solid Geometry
17.1 Solids
A solid is an enclosed portion of space bounded by plane and curved surfaces. Thus, the pyramid , the cube , the cone , the cylinder , and the sphere are solids. A solid has three dimensions: length, width, and thickness. Practical illustrations of solids include a box, a brick, a block, and a ball. These are not, however, the pure or ideal solids that are the concern of solid geometry. In solid geometry we study the geometric properties of perfect solids, such as their shape, their size, the relationships of their parts, and the relationships along solids; physical properties such as their color, weight, or smoothness are disregarded.
17.1A Kinds of Solids
Polyhedra A polyhedron is a solid bounded by plane (flat) surfaces only. Thus, the pyramid and cube are polyhedrons. The cone, cylinder, and sphere are not polyhedra, since each has a curved surface. The bounding surfaces of the polyhedron are its faces; the lines of intersection of the faces are its edges, and the points of intersection of its edges are its vertices. A diagonal of a polyhedron joins two vertices not in the same face. Thus, the polyhedron shown in Fig. 17-1 has six faces. Two of them are triangles (ABC and CFG), and the other four are quadrilaterals. Note AF is a diagonal of the polyhedron. The shaded polygon HJKL is a section of the polyhedron formed by the intersection of the solid and a plane passing through it.
Fig. 17-1
The angle between any two intersecting faces is a dihedral angle. The angle between the covers of an open book is a dihedral angle. As the book is opened wider, the dihedral angle grows from one that is acute to one that is right, obtuse, and then straight. The dihedral angle can be measured by measuring the plane angle between two lines, one in each face, and in a plane perpendicular to the intersection between the faces.
CHAPTER 17 Extending Plane Geometry into Solid Geometry
Prisms
A prism (Fig. 17-2) is a polyhedron two of whose faces are parallel polygons, and whose remaining faces are parallelograms. The bases of a prism are its parallel polygons. These may have any number of sides. The lateral (sides) faces are the parallelograms. The distance between the two bases is h; it is measured along a line at right angles to both bases. A right prism is a prism whose lateral faces are rectangles. The distance h is the height of any of the lateral faces.
Fig. 17-2
Fig. 17-3
A rectangular solid (box) is a prism bounded by six rectangles. The solid can be formed from a pattern of six rectangles, as shown in Fig. 17-3, folded along the dashed lines. The length l, width w, and height h are its dimensions. A cube is a rectangular solid bounded by six squares. The cube can be formed from a pattern of six squares, as shown in Fig. 17-4, folded along the dashed lines. Each equal dimension is represented by e in the diagram.
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