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-11811001270000350322-184068Part 1- Circles00Part 1- CirclesCircles - Cheat SheetTopic Overview49091608566800Center Radius Form, for circle centered at (h,k) with radius, r(x-h)2+(y-k)2=r2Completing the square for center-radius form1. Move loose numbers to one side2. Group x’s and y’s3. Divide middle term by 2 and square it – ADD TO BOTH SIDES!4. put factors into Squared Form 2 ( remember the number will be half of the middle term)5. You’re in center-radius form!!!We complete the square twice to put general form equations of circles into Center-Radius form, then graph!Recognize a circle by finding an x2Systems with CirclesAny point of intersection is a solution to the system – solve graphically!Systems with ParabolasAny point of intersection is a solution to the system – solve graphically!Watch out for sneaky turning points and sneaky solutions- know how to manipulate your calculator!( 12-6 was a quiz!)583405877676Formulas for area and circumference are on the reference sheet.00Formulas for area and circumference are on the reference sheet.Area of a Sector300157-8997KNOW DIFFERENCE B/T AREA and LENGTH (CIRCUMFERENCE)KNOW DIFFERENCE B/T AREA and LENGTH (CIRCUMFERENCE)Area of a Sector=Area of a Circleangle measure of sector360Area of a circle = πr2Arc Length of a Sector (In degrees)Arc Length=Circumferenceangle measure of sector360Circumference of a circle = πdSolving for arc length IN RADIANSs = rθ where: s = arc length; r = radius; θ = central angleRadians – unit of angle measureAn angle is 1 radian when the length of the arc of the circle is equal to the radiusConversionsSet up a proportion and solve for desired angle measure!radiansdegrees= π180Proving Circles SimilarTranslate to get centers to coincideDilate with center of dilation at one of the centers and scale factor using the radii.Circle Regents Type questions1. The equation of a circle is . What are the coordinates of the center and the length of the radius of the circle?1)center and radius 42)center and radius 43)center and radius 164)center and radius 162. A circle with a radius of 5 was divided into 24 congruent sectors. The sectors were then rearranged, as shown in the diagram below.To the nearest integer, the value of x is1)312)163)124)103. In the diagram below of circle O, the area of the shaded sector AOC is and the length of is 6 inches. Determine and state .YOU TRY!4. If is the equation of a circle, the length of the radius is1)252)163)54)45. Triangle FGH is inscribed in circle O, the length of radius is 6, and .What is the area of the sector formed by angle FOH?1)2)3)4)6. In the diagram below of circle O, the area of the shaded sector LOM is .If the length of is 6 cm, what is ?1)10?2)20?3)40?4)80?Circle THEOREMS - Cheat SheetCENTRAL ANGLESVERTEX MUST BE ON THE CENTER OF THE CIRCLE.97028062992000RULE: ANGLE = INTERCEPTED ARCNSCRIBED ANGLESVERTEX MUST BE ON THE CIRCLE.1727200264160Thales Theorem00Thales Theorem16802104489450036068032448500RULE: ANGLE =HALF THE INTERCEPTED ARCANGLES FORMED BY 2 CHORDS“BOW –TIE ANGLES”VERTEX is NOT on the center and NOT on the circle.RULE: ARC #1+ARC #22176593595313500Cyclic QuadrilateralsQuadrilateral inscribed in a circle27305354965001560195450850Watch out! This is not necessarily a parallelogram, so consecutive angles are not supplementary. ONLY OPPOSITE ANGLES.00Watch out! This is not necessarily a parallelogram, so consecutive angles are not supplementary. ONLY OPPOSITE ANGLES.OPPOSITE ANGLES ARE SUPPLEMETARY (ADD UP tO 180 )SPECIAL INSCRIBED ANGLES Formed by a Tangent and chord1749425211455001637030211455 SPECIAL CASEIf the chord is a diameter, rt angles!00 SPECIAL CASEIf the chord is a diameter, rt angles!Rule: ANGLE =HALF THE INTERCEPTED ARC9893307505700098933066421000-1905-635004559307429500Sneaky AngleFormed by a secant and chord80137035750500Rule: Find the measure of inscribed adjacent angle and subtract from 180.3463290147320Special Case- “ Ice cream Cone”Narc + Outside angle = 18000Special Case- “ Ice cream Cone”Narc + Outside angle = 180505333017843500ANGLES FORMED BY TWO SECANTS, OR TWO TANGENTS OR A SECANT AND A TANGENT.186944026543000RULE: Outside ?=FARC-NARC21) Always solve for arcs first. (HIGLIGHT DIAMETERS!).KNOW HOW TO WORK WITH RATIOS!2) Solve the parts in the order they are presented in for the problem.3) GO SLOW.4) Highlight parts you're solving for (in different colors).5) Fill in/mark up the diagramParallel ChordsThe arcs between two parallel chords are congruentCongruent ChordsThe arcs outside (subtended by) congruent chords are congruentSegment Length INSIDE Circle (PP)1085215351790part x part = part x partorpp = pp00part x part = part x partorpp = ppIf a diameter/radius is perpendicular to a chord, then it bisects that chord.8075401724100219531203960whole x outer = whole x outer orwo = wo00whole x outer = whole x outer orwo = woSegment Length OUTSIDE Circle A right angle is formed at the point of tangency between a tangent and diameter/radiusTangents that meet at one exterior point are congruent:7340601054103 common tangents003 common tangents6019805111754 common tangents004 common tangentsBig Circles (Yesterday’s Review)!Similar Circles1. Translate (if necessary) first - STATE VECTOR2. Dilate at a center of dilation and a scale factor (image / pre-image)3. ConcludeCircle Proofs TipsMark up your diagramCome up with a planUse your proof piecesLook for congruent angles and congruent sidesAlways look for radii (always congruent) and Inscribed Angles!!!Circle Regents Type questions1. In circle O shown below, diameter is perpendicular to at point C, and chords , , , and are drawn.Which statement is not always true?1)2)3)4)22. In the diagram shown below, is tangent to circle O at A and to circle P at C, intersects at B, , , and .What is the length of ?1)6.42)83)12.54)163. In the diagram below, quadrilateral ABCD is inscribed in circle P.What is ?1)70°2)72°3)108°4)110° You try!4. In the diagram below, , , , , and are chords of circle O, is tangent at point D, and radius is drawn. Sam decides to apply this theorem to the diagram: “An angle inscribed in a semi-circle is a right angle.”29508452730500Which angle is Sam referring to?1)2)3)4)In the diagram below of circle O with diameter and radius , chord is parallel to chord .If , determine and state .Part 2- QuadrilateralsQuadrilateral Properties Cheat Sheet – Sides and DiagonalsParallelogram Properties2 sets of opposite sides are parallel 2 sets of opposite sides are congruentDiagonals bisect each otherOpposite angles are congruent, Consecutive angles are supplementaryRectangle Properties2 sets of opposite sides are parallel2 sets of opposite sides are congruentDiagonals bisect each other*Diagonals are congruent*Adjacent sides are perpendicularRhombus Properties2 sets of opposite sides are parallel2 sets of opposite sides are congruentDiagonals bisect each other*All sides are congruent*Diagonals are perpendicularDiagonals bisect vertex anglesSquare Properties2 sets of opposite sides are parallel2 sets of opposite sides are congruentDiagonals bisect each otherDiagonals are congruentAdjacent sides are perpendicularAll sides are congruentDiagonals are perpendicularTrapezoid Properties541591554610Helps prove:00Helps prove:ONLY 1 set of opposite sides are parallelAngles between bases are supplentaryIsosceles Trapezoid PropertiesONLY 1 set of opposite sides are parallel*ONLY 1 set of opposite sides are congruent (legs)*Diagonals are congruent, BASE ANGLES ARE CONGRUENTTools:Slope:1171575175895Helps prove:00Helps prove:m= y2-y1x2-x1 or counting methodMidpoint: 826770327660Helps prove:00Helps prove:M=(x1+x22,y1+y22)Distance:d= (x2-x1)2+(y2-y1)2Quadrilateral Family “Geome-tree”11171581801771-15784751610376-15226751668696-17372351580136-1613035-1920279Another way to think about it….11493511620500 How you read this: All rectangles, rhombuses, and squares are parallelograms, but not all parallelograms are rectangles, squares and/or rhombuses. All squares are rectangles but not all rectangles are squares etc….Types of Quadrilateral Regents Questions:Property Multiple Choice QuestionsUsing a property to solveCoordinate Geometry ProofTwo-Column Quadrilateral ProofRegents type Questions1. Quadrilateral ABCD has diagonals and . Which information is not sufficient to prove ABCD is a parallelogram?1) and bisect each other.2) and 3) and 4) and 2. The diagram below shows parallelogram LMNO with diagonal , , and .Explain why is 40 degrees.3. 1) Coordinate Geometry Proof2) Which statement is not always true about a rhombus? 1)The diagonals are perpendicular2)The opposite sides are congruent.3)The adjacent sides are perpendicular.4)The opposite sides are parallel.3) In the accompanying diagram of rhombus ABCD, Sides AB and BC are adjacent sides, If AB=4x , and 517906017018000 BC=x+3What is the value of x? What is the length of AB? 1) In the diagram below, quadrilateral STAR is a rhombus with diagonals and intersecting at E. , , , , , , and . 4143375-81280a) Solve for SR b) Solve for RTc) Solve for2) Given: ABCD is a parallelogram. ?ADB ? ?BCA. Prove ABCD is a rectangle.38100908050Solids and 3D ShapesKey ConceptsImportant Notes14-1: Calculating Area of Regular Polygons34194756540500Need to “break up the figure” into trianglesSteps:Calculate the apothem!4382135101663500 If not already there, draw in the Apothem (mark the right angles) and bisect the central angleto find the vertex angle of the small RIGHT triangle.Bisect the base of the isosceles triangle to find the length of one side of the right triangle. Use SOH CAH TOA to calculate the length of the apothem.Find the area of each triangleMultiply the area by the # of triangles in the regular polygon. 14-2: Translations forming Solids, Properties and cross sections4635500-1016000CylindersFormed by translation Base shape: circle (parallel and congruent shapes in the 3D figure)Lateral View: Rectangle (cross section perpendicular to the base)Base View: Circle (cross section parallel to the base)Prisms46069254318000Formed by translationPolygonal basesNamed by their base shapesLateral View: Rectangle (cross section perpendicular to the base)Base View: same as base shape (cross section parallel to the base)279717512954000PyramidFormed by translation and dilationOne base that is a polygonNamed by base shapeLateral edges congruent14-3: Rotations forming Solids, Properties and cross sections45497755016500*Cylinders can be formed by rotationsConeFormed by rotation of triangle Slant height: height from edge of base to topBase shape: circle (parallel and congruent shapes in the 3D figure)456057015621000Lateral View: Triangle (cross section perpendicular to the base)Base View: Circle (cross section parallel to the base)SphereFormed by rotation of circle or semi-cirlce Lateral View: Circle (cross section perpendicular to the base)Base View: Circle (cross section parallel to the base)The great circle: largest circle within a sphere; same diameter as sphere14-4: Volume of Prisms and Cylinders, equal Volumes, Cavalieri’s PrincipleVolume of Prisms and Cylinders:V = Bh(B = area of the base; h is the height/depth of the prism/distance between the bases)4073525825500Cavalieri’s Principle: two of the same solid that have thesame base area and the same height, also have the same volume.14-5: Volume of Pyramids, Cones and SpheresV(pyramid/cone) = 13BhV(sphere) = 43πr314-6: Density and Surface AReaPopulation DensityA ratio of the amount of a population that exists over a given areaPopulation Density = populationtotal areaDensity of a 3D SolidA ratio that compares an object’s weight (mass) to the amount of space (volume) it takes upDensity = MassVolume3 Dimensional Solids PropertiesFeatures:PrismCylinderConePyramidsSphereBase ViewAlways polygonCircleCircleAny PolygonCircleLateral ViewRectangleRectangleIsosceles TriangleIsosceles TriangleCircleFormationTranslating a polygon into 3 DimensionsTranslating a circle into 3 DimensionsOrRotating a RectangleRotating a TriangleTranslation and Dilation of a polygonRotate a semi-circleCross Section Parallel to the BaseSame 2D shape as base viewCross Section Perpendicular to the BaseSame 2D shape as lateral viewNumber of Bases2211n/aExample12954012382500Formulas (Volume)V = (area of the base) height V = (area of the base) heightV =13 (area of the base) heightV =13 (area of the base) heightHelpful Tips For SolidsBe prepared for application problemsCareful typing into your calculator (especially fractions)Don’t Round until the end of the problemLeave in terms of Pi until the end of the problem (sometimes pi will cancel)Use appropriate formulasIf you see equal height or equal volume, that’s Cavalieri’s Principle!Be prepared to describe how solids are formed through transformationBe prepared to describe what cross sections look likeSOLIDS PRACICEWhen a circle is translated a __________________________ is the solid formedWhen a cylinder is cut by a plane, parallel to it’s base, the cross section is a ____________________All cross sections of a sphere are circles a) Trueb) FalseWhen a right triangle is rotated, the solid formed is a _______________________Find the volume of the following right circular cylinder. Round to the nearest tenth.57232556584315The volume of a right circular cone is 80π and a radius of 4. Find the altitude.The radius of a sphere is 2 feet. Find the volume of the sphere in terms of π.4. In a solid hemisphere, a cone is removed as shown. Calculate the volume of the resulting solid to the nearest hunderedth. In addition to your solution, provide an explanation of the strategy you used in your solution.50399954445000 Find the area of a regular nonagon below.Calculate the Surface area and volume of the following solid:a) The surface area of the prism below is 102 cm2. Find x.b) If 1 can of paint covers 30cm2, how many cans of paint do you need to buy to cover the prism above?The water tower in the picture below is modeled by the two-dimensional figure beside it. The water tower is composed of a hemisphere, a cylinder, and a cone. Let C be the center of the hemisphere and let D be the center of the base of the cone.47529756604000 If feet, feet, and , determine and state, to the nearest cubic foot, the volume of the water tower. Two containers show below hold candies of the same size. Container A holds 75 candies and container B holds 160 candies. Given the dimensions below, which container has a smaller population density? A contractor needs to purchase 500 bricks. The dimensions of each brick are 5.1 cm by 10.2 cm by 20.3 cm, and the density of each brick is . The maximum capacity of the contractor’s trailer is 900 kg. Can the trailer hold the weight of 500 bricks? Justify your answer.a) Notice how density is given in kilograms per meters cubed? 1st convert dimensions of the brick so it’s in meters too!b) Now volume of a brick.c) Fill into density formula; find the mass (weight) of one brick.d) Can the trailer hold 500 bricks? Justify your answer! ................
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