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8.1 Prisms, Area and Volume Prism – 2 congruent polygons lie in parallel planes corresponding sides are parallel. corresponding vertices are connected base edges are edges of the polygons lateral edges are segments connecting corresponding vertices

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8.1 Prisms, Area and Volume Right Prism – prism in which the lateral edges are to the base edges at their points of intersection. Oblique Prism – Lateral edges are not perpendicular to the base edges. Lateral Area (L) – sum of areas of lateral faces (sides).

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**8.1 Prisms, Area and Volume Lateral area of a right prism: L = hP**

h = height (altitude) of the prism P = perimeter of the base (use perimeter formulas from chapter 7) Total area of a right prism: T = 2B + L B = base area of the prism (use area formulas from chapter 7)

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8.1 Prisms, Area and Volume Volume of a right rectangular prism (box) is given by V = lwh where l = length w = width h = height l w h

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**8.1 Prisms, Area and Volume Volume of a right prism is given by V = Bh**

B = area of the base (use area formulas from chapter 7) h = height (altitude) of the prism

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**8.2 Pyramids Area, and Volume**

Regular Pyramid – pyramid whose base is a regular polygon and whose lateral edges are congruent. Triangular pyramid: base is a triangle Square pyramid: base is a square

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**8.2 Pyramids Area, and Volume**

Slant height (l) of a pyramid: The altitude of the congruent lateral faces. Slant height height apothem

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**8.2 Pyramids Area, and Volume**

Lateral area - regular pyramid with slant height = l and perimeter P of the base is: L = ½ lP Total area (T) of a pyramid with lateral area L and base area B is: T = L + B Volume (V) of a pyramid having a base area B and an altitude h is:

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8.3 Cylinders and Cones Right circular cylinder: 2 circles in parallel planes are connected at corresponding points. The segment connecting the centers is to both planes.

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8.3 Cylinders and Cones Lateral area (L) of a right cylinder with altitude of height h and circumference C Total area (T) - cylinder with base area B Volume (V) of a cylinder is V = B h

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8.3 Cylinders and Cones Right circular cone – if the axis which connects the vertex to the center of the base circle is to the plane of the circle.

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8.3 Cylinders and Cones In a right circular cone with radius r, altitude h, and slant height l (joins vertex to point on the circle), l2 = r2 + h2 Slant height height radius

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8.3 Cylinders and Cones Lateral area (L) of a right circular cone is: L = ½ lC = rl where l = slant height Total area (T) of a cone: T = B + L (B = base circle area = r2) Volume (V) of a cone is:

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**8.4 Polyhedrons and Spheres**

Polyhedron – is a solid bounded by plane regions. A prism and a pyramid are examples of polyhedrons Euler’s equation for any polyhedron: V+F = E+2 V - number of vertices F - number of faces E - number of edges

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**8.4 Polyhedrons and Spheres**

Regular Polyhedron – is a convex polyhedron whose faces are congruent polygons arranged in such a way that adjacent faces form congruent dihedral angles. tetrahedron

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**8.4 Polyhedrons and Spheres**

Examples of polyhedrons (see book) Tetrahedron (4 triangles) Hexahedron (cube – 6 squares) Octahedron (8 triangles) Dodecahedron (12 pentagons)

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**8.4 Polyhedrons and Spheres**

Sphere formulas: Total surface area (T) = 4r2 Volume radius

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**9.1 The Rectangular Coordinate System**

Distance Formula: The distance between 2 points (x1, y1) and (x2,y2) is given by the formula: What theorem in geometry does this come from?

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**9.1 The Rectangular Coordinate System**

Midpoint Formula: The midpoint M of the line segment joining (x1, y1) and (x2,y2) is : Linear Equation: Ax + By = C (standard form)

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**9.2 Graphs of Linear Equations and Slopes**

Slope – The slope of a line that contains the points (x1, y1) and (x2,y2) is given by: rise run

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**9.2 Graphs of Linear Equations and Slopes**

If l1 is parallel to l2 then m1 = m2 If l1 is perpendicular to l2 then: (m1 and m2 are negative reciprocals of each other) Horizontal lines are perpendicular to vertical lines

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**9.3 Preparing to do Analytic Proofs**

To prove: You need to show: 2 lines are parallel m1 = m2, using 2 lines are perpendicular m1 m2 = -1 2 line segments are congruent lengths are the same, using A point is a midpoint

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**9.3 Preparing to do Analytic Proofs**

Drawing considerations: Use variables as coordinates, not (2,3) Drawing must satisfy conditions of the proof Make it as simple as possible without losing generality (use zero values, x/y-axis, etc.) Using the conclusion: Verify everything in the conclusion Use the right formula for the proof

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9.4 Analytic Proofs Analytic proof – A proof of a geometric theorem using algebraic formulas such as midpoint, slope, or distance Analytic proofs pick a diagram with coordinates that are appropriate. decide on what formulas needed to reach conclusion.

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**9.4 Analytic Proofs Triangles to be used for proofs are in: table 9.1**

Quadrilaterals to be used for proofs are in: table 9.2. The diagram for an analytic proof test problem will be given on the test.

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**9.5 Equations of Lines General (standard) form: Ax + By = C**

Slope-intercept form: y = mx + b (where m = slope and b = y-intercept) Point-slope form: The line with slope m going through point (x1, y1) has the equation: y – y1 = m(x – x1)

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9.5 Equations of Lines Example: Find the equation in slope-intercept form of a line passing through the point (-4,5) and perpendicular to the line 2x+3y=6 (solve for y to get slope of line) (take the negative reciprocal to get the slope)

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9.5 Equations of Lines Example (continued): Use the point-slope form with this slope and the point (-4,5) In slope intercept form:

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