ssrs barcode font free Extending Plane Geometry into Solid Geometry in Objective-C

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CHAPTER 17 Extending Plane Geometry into Solid Geometry
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Locus in a Plane 2. The locus of points at a given distance from a given line is a pair of lines parallel to the given line and at the given distance from it.
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Locus in Space 2. The locus of points at a given distance from a given plane is a pair of planes parallel to the given plane and at the given distance from it.
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3. The locus of points equidistant from two points is the line that is the perpendicular bisector of the segment joining the two points.
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3. The locus of points equidistant from two points is the plane that is the perpendicular bisector of the segment joining the two points.
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4. The locus of points equidistant from the lines that are the sides of an angle is the line which is the bisector of the angle between them.
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4. The locus of points equidistant from the planes that are the sides of a dihedral angle is the plane which is the bisector of the angle between them.
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5. The locus of points equidistant from two parallel lines is the line parallel to them and midway between them.
5. The locus of points equidistant from two parallel planes is the plane parallel to them and midway between them.
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6. The locus of points equidistant from two concentric circles is the circle midway between the given circles and concentric to them.
6. The locus of points equidistant from two concentric spheres is the sphere midway between the given spheres and concentric to them.
CHAPTER 17 Extending Plane Geometry into Solid Geometry
Locus in a Plane 7. The locus of the vertex of a right triangle having a given hypotenuse is the circle having the hypotenuse as its diameter.
Locus in Space 7. The locus of the vertex of a right triangle having a given hypotenuse is the sphere having the hypotenuse as its diameter.
17.2B Extension of Analytic Geometry to Three-Dimensional Space
The analytic (or coordinate) geometry of two dimensions can easily be extended to three dimensions. Figure 17-13 shows the two- and three-dimensional axes; the z-axis is perpendicular to both the x- and y-axes. In the figure, arrows indicate the positive directions, and dashed lines indicate the negative axes. Four specific extensions from geometry in a plane to geometry in space follow.
z y O
O y
Fig. 17-13
Coordinate Geometry in a Plane 1. Coordinates: P1 (x1, y1), P2(x2, y2) 2. Midpoint of P1P2: x1 x2 y1 y2 , yM xM 2 2 3. Distance P1P2: d 2(x2 x1)2 (y2 y1)2 4. Equation of a circle having the origin as center and radius r: x2 y2 r2
Coordinate Geometry in Space 1. Coordinates: P1 (x1, y1, z1), P2 (x2, y2, z2) 2. Midpoint of P1P2: x1 x2 y1 y2 z1 z2 , yM , zM xM 2 2 2 3. Distance P1P2: d 2(x2 x1)2 (y2 y1)2 (z2 z1)2 4. Equation of a sphere having the origin as center and radius r: x2 y2 z2 r2
17.3 Areas of Solids: Square Measure
The area of each face of the cube in Fig. 17-14 is A S e2. The total surface area S of the cube is then 6e2
CHAPTER 17 Extending Plane Geometry into Solid Geometry
The areas of the six rectangles that make up the rectangular solid in Fig 17-15 are A A A lw lh for top and bottom faces for front and back faces
wh for left and right faces
The total surface area S of the rectangular solid is then S 2lw 2lh 4 pr2 2prh 2pr(r h)
r e e e h l w h r
The total surface area S of the sphere in Fig. 17-16 is S
The total surface area S of the right circular cylinder in Fig. 17-17 is S 2pr2
Fig 17-14
Fig. 17-15
Fig. 17-16
Fig 17-17
SOLVED PROBLEM
Finding total surface areas of solids Find, to the nearest integer, the total surface area of (a) A cube with an edge of 5 m (Fig. 17-18) (b) A rectangular solid with dimensions of 10 ft, 7 ft, and 42ft (Fig. 17-19) (c) A sphere with a radius of 1.1 cm (Fig. 17-20)
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