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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.

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Presentation on theme: "8.1 Prisms, Area and Volume Prism – 2 congruent polygons lie in parallel planes –corresponding sides are parallel. –corresponding vertices are connected."— Presentation transcript:

1 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

2 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).

3 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)

4 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 h w l

5 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

6 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

7 8.2 Pyramids Area, and Volume Slant height (l) of a pyramid: The altitude of the congruent lateral faces. height apothem Slant height

8 8.2 Pyramids Area, and Volume Lateral area - regular pyramid with slant height = l and perimeter P of the base is: L = ½ l  P 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:

9 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.

10 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

11 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.

12 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), l 2 = r 2 + h 2 height radius Slant height

13 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 =  r 2 ) Volume (V) of a cone is:

14 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

15 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

16 8.4 Polyhedrons and Spheres Examples of polyhedrons (see book) –Tetrahedron (4 triangles) –Hexahedron (cube – 6 squares) –Octahedron (8 triangles) –Dodecahedron (12 pentagons)

17 8.4 Polyhedrons and Spheres Sphere formulas: –Total surface area (T) = 4  r 2 –Volume radius

18 9.1 The Rectangular Coordinate System Distance Formula: The distance between 2 points (x 1, y 1 ) and (x 2,y 2 ) is given by the formula: What theorem in geometry does this come from?

19 9.1 The Rectangular Coordinate System Midpoint Formula: The midpoint M of the line segment joining (x 1, y 1 ) and (x 2,y 2 ) is : Linear Equation: Ax + By = C (standard form)

20 9.2 Graphs of Linear Equations and Slopes Slope – The slope of a line that contains the points (x 1, y 1 ) and (x 2,y 2 ) is given by: rise run

21 9.2 Graphs of Linear Equations and Slopes If l 1 is parallel to l 2 then m 1 = m 2 If l 1 is perpendicular to l 2 then: (m 1 and m 2 are negative reciprocals of each other) Horizontal lines are perpendicular to vertical lines

22 9.3 Preparing to do Analytic Proofs To prove:You need to show: 2 lines are parallelm 1 = m 2, using 2 lines are perpendicular m 1  m 2 = -1 2 line segments are congruentlengths are the same, using A point is a midpoint

23 9.3 Preparing to do Analytic Proofs Drawing considerations: 1.Use variables as coordinates, not (2,3) 2.Drawing must satisfy conditions of the proof 3.Make it as simple as possible without losing generality (use zero values, x/y-axis, etc.) Using the conclusion: 1.Verify everything in the conclusion 2.Use the right formula for the proof

24 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.

25 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.

26 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 (x 1, y 1 ) has the equation: y – y 1 = m(x – x 1 )

27 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)

28 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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