MATH 2160 3 rd Exam Review

MATH 2160 3MATH 2160 3rdrd Exam Exam ReviewReview

GeometryGeometry

and and

MeasurementMeasurement

Problem Solving – Problem Solving – Polya’s 4 StepsPolya’s 4 Steps

Understand the problemUnderstand the problem What does this mean?What does this mean? How do you understand?How do you understand?

Devise a planDevise a plan What goes into this step?What goes into this step? Why is it important?Why is it important?

Carry out the planCarry out the plan What happens here?What happens here? What belongs in this step?What belongs in this step?

Look backLook back What does this step imply?What does this step imply? How do you show you did this?How do you show you did this?

Problem SolvingProblem Solving Polya’s 4 StepsPolya’s 4 Steps

Understand the problemUnderstand the problem Devise a planDevise a plan Carry out the problemCarry out the problem Look backLook back

Which step is most important?Which step is most important? Why is the order important?Why is the order important? How has learning problem solving How has learning problem solving

skills helped you in this or another skills helped you in this or another course?course?

Problem Solving Problem Solving Strategies Strategies Make a chartMake a chart Make a tableMake a table Draw a pictureDraw a picture Draw a diagramDraw a diagram Guess, test, and reviseGuess, test, and revise Form an algebraic modelForm an algebraic model Look for a patternLook for a pattern Try a simpler version of the problemTry a simpler version of the problem Work backwardWork backward Restate the problem differentlyRestate the problem differently Eliminate impossible situationsEliminate impossible situations Use reasoningUse reasoning

GeometryGeometry

Angles and congruencyAngles and congruency CongruentCongruent– same size, same shape– same size, same shape Degree measureDegree measure – real number – real number

between 0 and 360 degrees that defines between 0 and 360 degrees that defines the amount of rotation or size of an anglethe amount of rotation or size of an angle

Sum of the interior anglesSum of the interior angles of any of any polygon: polygon: (n – 2)180(n – 2)180oo where n is the where n is the number of sides in the polygonnumber of sides in the polygon

GeometryGeometry

Special anglesSpecial angles right angleright angle – 90 – 90 acute angleacute angle – 0 – 0< angle < 90< angle < 90 obtuse angleobtuse angle – 90 – 90< angle < 180< angle < 180

Sum of the anglesSum of the angles Triangle = 180Triangle = 180oo

Quadrilateral = 360Quadrilateral = 360oo

Pentagon = 540Pentagon = 540oo

Etc. Etc.

GeometryGeometry CirclesCircles

circle circle – special simple closed curve – special simple closed curve where all points in the curve are where all points in the curve are equidistant from a given point in the equidistant from a given point in the same plane – same plane – NOTENOTE: Circles are : Circles are NOTNOT polygons!polygons!

center center – middle point of the circle– middle point of the circle diameter diameter – is a chord that passes – is a chord that passes

through the center of the circlethrough the center of the circle radius radius – line segment connecting the – line segment connecting the

center of the circle to any point on the center of the circle to any point on the circlecircle

GeometryGeometry Polygons – made up of line Polygons – made up of line

segmentssegments Triangles Triangles – 3-sided polygons– 3-sided polygons QuadrilateralsQuadrilaterals – 4-sided polygons – 4-sided polygons n - gons n - gons – the whole number n – the whole number n

represents the number of sides for the represents the number of sides for the polygon: a triangle is a 3-gon; a polygon: a triangle is a 3-gon; a square is a 4-gon; and so onsquare is a 4-gon; and so on

Regular PolygonsRegular Polygons – polygon where – polygon where the all the line segments and all of the the all the line segments and all of the angles are congruentangles are congruent

GeometryGeometry TrianglesTriangles

Union of three line segments formed Union of three line segments formed by three distinct non-collinear pointsby three distinct non-collinear points verticesvertices – intersection points of line – intersection points of line

segments forming the angles of the segments forming the angles of the polygonpolygon

sidessides – the line segments forming the – the line segments forming the polygonpolygon

height height – line segment from a vertex of a – line segment from a vertex of a triangle to a line containing the side of the triangle to a line containing the side of the triangle opposite the vertex triangle opposite the vertex

GeometryGeometry TrianglesTriangles

equilateralequilateral – all sides and angles – all sides and angles congruent congruent

isoscelesisosceles – at least one pair of – at least one pair of congruent sides and anglescongruent sides and angles

scalenescalene – no congruent sides or – no congruent sides or anglesangles

rightright – one right angle – one right angle acuteacute – all angles acute – all angles acute obtuseobtuse – one obtuse angle – one obtuse angle

GeometryGeometry QuadrilateralsQuadrilaterals

parallelogramparallelogram – quadrilateral with two – quadrilateral with two pairs of parallel sidespairs of parallel sides opposite sides are parallelopposite sides are parallel opposite sides are congruentopposite sides are congruent

rectanglerectangle – quadrilateral with four – quadrilateral with four right anglesright angles a a parallelogram is a rectangleparallelogram is a rectangle if and if and

only ifonly if it has at least one right angleit has at least one right angle

trapezoidtrapezoid – exactly one pair of – exactly one pair of opposite sides parallel, but not opposite sides parallel, but not congruentcongruent

GeometryGeometry QuadrilateralsQuadrilaterals

rhombusrhombus – quadrilateral with four – quadrilateral with four congruent sidescongruent sides a a parallelogram is a rhombusparallelogram is a rhombus if and if and

only ifonly if it has four congruent sidesit has four congruent sides

squaresquare – quadrilateral with four right – quadrilateral with four right angles and four congruent sidesangles and four congruent sides a a square is a parallelogramsquare is a parallelogram if and only if if and only if it is a rectangle with four congruent sidesit is a rectangle with four congruent sides it is a rhombus with a right angleit is a rhombus with a right angle

GeometryGeometry PentominosPentominos

* Won't fold into an open box* Won't fold into an open box

f p*

j n

i*

GeometryGeometry

PentominosPentominos * Won't fold into an open box* Won't fold into an open box

t u* v*

GeometryGeometry

PentominosPentominos

w x z

y

GeometryGeometry

Patterns with points, lines, and Patterns with points, lines, and regionsregions Where k is the number of lines or line Where k is the number of lines or line

segmentssegments

P = [k (k – 1)] / 2P = [k (k – 1)] / 2 Regions = lines + points + 1Regions = lines + points + 1 R = k + P + 1 = [k (k + 1)] / 2 + 1R = k + P + 1 = [k (k + 1)] / 2 + 1

1k

1n

xsintPo

GeometryGeometry

TangramsTangrams Flips, slides, and turnsFlips, slides, and turns CommunicationCommunication MapsMaps Conservation of AreaConservation of Area

PiagetPiaget If use all of the pieces to make a new If use all of the pieces to make a new

shape, both shapes have the same shape, both shapes have the same areaarea

GeometryGeometry PolyhedronPolyhedron

VerticesVertices EdgesEdges FacesFaces

Should be able to draw Should be able to draw ALLALL of the of the following:following: SphereSphere Prisms – Cube, Rectangular, Prisms – Cube, Rectangular,

TriangularTriangular CylinderCylinder ConeCone Pyramids – Triangular, SquarePyramids – Triangular, Square


RectangleRectangle Perimeter Perimeter P = 2l + 2w, where l = length and P = 2l + 2w, where l = length and

w = widthw = width Example: l = 5 ft and w = 3 ftExample: l = 5 ft and w = 3 ft

PP rectangle rectangle == 2l + 2w2l + 2w

PP == 2(5 ft) + 2(3 ft)2(5 ft) + 2(3 ft) PP == 10 ft + 6 ft10 ft + 6 ft PP == 16 ft16 ft

3 ft

5 ft


RectangleRectangle Area Area A = lw where l = length and w = A = lw where l = length and w =

widthwidth Example: l = 5 ft and w = 3 ftExample: l = 5 ft and w = 3 ft

AA rectangle rectangle = = lwlw

AA == (5 ft)(3 ft)(5 ft)(3 ft) A A == 15 ft15 ft22

3 ft

5 ft


SquareSquare Perimeter Perimeter P = 4s, where s = length of a sideP = 4s, where s = length of a side Example: s = 3 ftExample: s = 3 ft

PP square square == 4s4s

PP == 4(3 ft)4(3 ft) PP == 12 ft12 ft

3 ft


SquareSquare Area Area A = sA = s22 where s = length of a side where s = length of a side Example: s = 3 ftExample: s = 3 ft

AA square square = = ss22

AA == (3 ft)(3 ft)22

A A == 9 ft9 ft22

3 ft


TriangleTriangle PerimeterPerimeter PP = a + b + c, where a, b, and c = a + b + c, where a, b, and c

are the lengths of the sides of the are the lengths of the sides of the triangletriangle

Example: a = 3 m; b = 4 m; c = 5 Example: a = 3 m; b = 4 m; c = 5 mm PP triangle triangle == a + b + ca + b + c PP == 3 m + 4 m + 5 m3 m + 4 m + 5 m PP == 12 m12 m

3 m

5 m4 m


TriangleTriangle AreaArea A = ½ bh, where b is the base and A = ½ bh, where b is the base and

h is the height of the triangleh is the height of the triangle Example: b = 3 m; h = 4 mExample: b = 3 m; h = 4 m

AA triangle triangle == ½ bh½ bh AA == ½ (3 m) (4 m)½ (3 m) (4 m) AA == 6 m6 m22

3 m

5 m

4 m


CircleCircle CircumferenceCircumference

CC circle circle = = d or C = 2d or C = 2r, where d = r, where d =

diameter and r = radiusdiameter and r = radius Example: r = 3 cmExample: r = 3 cm

CC circle circle == 2 2rr

CC == 2 2(3 cm)(3 cm) CC == 6 6 cm cm

3 cm


CircleCircle AreaArea A = A = rr22, where r = radius, where r = radius Example: r = 3 cmExample: r = 3 cm

A A circlecircle == rr22

AA == (3 cm)(3 cm)22

AA == 9 9 cm cm22

3 cm


Rectangular PrismRectangular Prism Surface AreaSurface Area: sum of the areas of all of the : sum of the areas of all of the

facesfaces ExampleExample: There are 4 lateral faces: 2 lateral : There are 4 lateral faces: 2 lateral

faces are 6 cm by 7 cm (Afaces are 6 cm by 7 cm (A11= wh) and 2 = wh) and 2 lateral faces are 5 cm by 7 cm (Alateral faces are 5 cm by 7 cm (A22 = lh). = lh). There are 2 bases 6 cm by 5 cm (AThere are 2 bases 6 cm by 5 cm (A33 = lw) = lw)

AA11 = (6 cm)(7 cm) = 42 cm = (6 cm)(7 cm) = 42 cm22

AA22 = (5 cm)(7 cm) = 35 cm = (5 cm)(7 cm) = 35 cm22

AA33 = (6 cm)(5 cm) = 30 cm = (6 cm)(5 cm) = 30 cm22

SA SA rectangular prismrectangular prism = 2wh + 2lh + 2lw = 2wh + 2lh + 2lw

SA = 2(42 cmSA = 2(42 cm22) + 2(35 cm) + 2(35 cm22) + 2(30 cm) + 2(30 cm22)) SA = 84 cmSA = 84 cm22 + 70 cm + 70 cm22 + 60 cm + 60 cm22

SA = 214 cmSA = 214 cm22

7 cm

6 cm

5 cm


Rectangular Prism Rectangular Prism VolumeVolume: : V = lwh where l is length; w is width; V = lwh where l is length; w is width;

and h is heightand h is height ExampleExample: l = 6 cm; w = 5 cm; h = 7 cm: l = 6 cm; w = 5 cm; h = 7 cm

V V rectangular prismrectangular prism = Bh = lwh = Bh = lwh

VV == (6 cm)(5 cm)(7 cm)(6 cm)(5 cm)(7 cm) VV == 210 cm210 cm33

7 cm

6 cm

5 cm


CubeCube Surface AreaSurface Area: sum of the areas of all : sum of the areas of all

6 congruent faces6 congruent faces ExampleExample: There are 6 faces: 5 cm by : There are 6 faces: 5 cm by

5 cm (A = s5 cm (A = s22))

SA SA cubecube = 6A = 6s = 6A = 6s22

SA = 6(5 cm)SA = 6(5 cm)22

SA = 6(25 cmSA = 6(25 cm22)) SA = 150 cmSA = 150 cm22

5 cm


Cube Cube VolumeVolume: : V = sV = s33 where s is the length of a side where s is the length of a side ExampleExample: s = 5 cm: s = 5 cm

V V cubecube = Bh = s= Bh = s33

VV == (5 cm)(5 cm)33

VV == 125 cm125 cm33

5 cm


Triangular PrismTriangular Prism Surface AreaSurface Area: sum of the areas of all of the : sum of the areas of all of the

facesfaces ExampleExample: There are 3 lateral faces: 6 m by : There are 3 lateral faces: 6 m by

7 m (A7 m (A11= bl). There are 2 bases: 6 m for the = bl). There are 2 bases: 6 m for the

base and 5 m for the height (2Abase and 5 m for the height (2A22 = bh). = bh). AA11 = (6 m)(7 m) = 42 m = (6 m)(7 m) = 42 m22

2A2A22 = (6 m)(5 m) = 30 m = (6 m)(5 m) = 30 m22

SA SA triangular prismtriangular prism = bh + 3bl = bh + 3bl

SA = 30 mSA = 30 m22 + 3(42 m + 3(42 m22)) SA = 30 mSA = 30 m22 + 126 m + 126 m22

SA = 156 mSA = 156 m22

7 m

6 m

5 m


Triangular Prism Triangular Prism VolumeVolume: : V = ½ bhl where b is the base; h is V = ½ bhl where b is the base; h is

height of the triangle; and l is length of height of the triangle; and l is length of the prismthe prism

ExampleExample: b = 6 m; h = 5 m; l = 7 m: b = 6 m; h = 5 m; l = 7 m

V V triangular prismtriangular prism = Bh = ½ bhl = Bh = ½ bhl

VV == ½ (6 m)(5 m)(7 m)½ (6 m)(5 m)(7 m) VV == 105 m105 m33

7 m

6 m

5 m


CylinderCylinder Surface AreaSurface Area: area of the circles plus : area of the circles plus

the area of the lateral facethe area of the lateral face ExampleExample: r = 3 ft; h = 12 ft: r = 3 ft; h = 12 ft

SA SA cylindercylinder= = 22rh +2rh +2rr22

SA = 2SA = 2 (3 ft)(12 ft) + 2 (3 ft)(12 ft) + 2 (3 ft) (3 ft)22 SA SA == 7272 ft ft22 + 2 + 2 (9 ft (9 ft22)) SASA == 7272 ft ft22 + 18 + 18 ft ft22

SASA = = 9090 ft ft22

3 ft

12 ft


CylinderCylinder Volume of a CylinderVolume of a Cylinder: V = : V = rr22h h

where r is the radius of the base where r is the radius of the base (circle) and h is the height.(circle) and h is the height.

ExampleExample: r = 3 ft and h = 12 ft.: r = 3 ft and h = 12 ft. VV cylinder cylinder == Bh = Bh = rr22hh VV == (3 ft)(3 ft)22 (12 ft) (12 ft) VV == (9 ft(9 ft22)(12 ft))(12 ft) VV == 108108 ft ft33

3 ft

12 ft


ConeCone Surface AreaSurface Area: area of the circle plus : area of the circle plus

the area of the lateral facethe area of the lateral face ExampleExample: r = 5 ft; t = 13 ft: r = 5 ft; t = 13 ft

SA SA conecone= = rt +rt +rr22

SA = SA = (5 ft)(13 ft) + (5 ft)(13 ft) + (5 ft) (5 ft)22 SA SA == 6565 ft ft22 + + (25 ft (25 ft22)) SASA == 6565 ft ft22 + 25 + 25 ft ft22

SASA = = 9090 ft ft22

5 ft

13 ft

12 ft


ConeCone VolumeVolume: V = : V = rr22h/3 where r is the h/3 where r is the

radius of the base (circle) and h is the radius of the base (circle) and h is the height.height.

ExampleExample: r = 5 ft; h = 12 ft: r = 5 ft; h = 12 ft V V conecone= = rr22h/3h/3 V V = = [[(5 ft)(5 ft)22 12 ft ]/ 3 12 ft ]/ 3 V V == [(25[(25 ft ft22)(12 ft)]/3)(12 ft)]/3 VV == (25(25 ft ft22)(4 ft))(4 ft) VV = = 100100 ft ft33

5 ft

13 ft

12 ft


SphereSphere Surface AreaSurface Area: 4: 4rr22 where r is the where r is the

radiusradius ExampleExample: r = 8 mm: r = 8 mm SA SA sphere sphere = = 44rr22

SASA = = 44(8 mm)(8 mm)22 SASA = = 44(64 mm(64 mm22)) SA SA = = 256256 mm mm22

8 mm


SphereSphere Volume of a SphereVolume of a Sphere: V = (4/3): V = (4/3) r r33

where r is the radiuswhere r is the radius ExampleExample: r = 6 mm: r = 6 mm V V spheresphere == 44rr33/3/3 VV == [4[4 x (6 mm) x (6 mm)33]/3]/3 VV == [4[4 x 216 mm x 216 mm33]/3]/3 VV == [864[864 mm mm33]/3]/3 VV == 288288 mm mm33

6 mm


Triangular PyramidTriangular Pyramid

Square PyramidSquare Pyramid

Test Taking TipsTest Taking Tips

Get a good nights rest before the Get a good nights rest before the examexam

Prepare materials for exam in Prepare materials for exam in advance (scratch paper, pencil, and advance (scratch paper, pencil, and calculator)calculator)

Read questions carefully and ask if Read questions carefully and ask if you have a question DURING the you have a question DURING the examexam

Remember: If you are prepared, you Remember: If you are prepared, you need not fearneed not fear

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MATH 2160 3 rd Exam Review