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Name:_____________________________________________________ ForStudentsEnteringGeometry I. Simplifyeachexpression.Showallwork. 1. 3 4 β 5 β 2! 3. 3 + 4(6) 2 5. ! 4π₯ ! 3π₯ ! 6. 5π β 2π β 3π β π 7. 3π₯ 6π₯ ! 8. 4. 18 2. 3 β 2 4 + 5 β 2 3π₯ ! ! II. Solveforeachvariable. 9. 4π₯ β 3 = 2(π₯ + 1) 11. SolveforH: π = πΏ β π β π» 10. π₯ + 4 3 = 6 7 12. Ifπ + π = 16,then3π + 3π = FORTHEFOLLOWINGSECTIONSPLEASEREADTHEGIVEN INFORMATIONBEFOREYOUWORKONTHEPRACTICEPROBLEMS. Radicals Tips: β’ You will need to know all your perfect squares 1! = 1 through 20! = 400. !"#$% β’ ππππππππ Radical symbol Guided Practice: Simplify Radicals Simplify: β5 128 β5 64 β 2 β split the radicand into a factor pair, one of which is a perfect square β5 β 64 β 2 β put each factor under its own radical symbol β5 β 8 β 2 β find the square root of the perfect square factor β40 2 β simplify Guided Practice: Add/Subtract Radicals Tips: β’ In order to add or subtract radicals, the index and the radicand must be the same! β’ You combine radicals as if you were combining like-termsβadd/subtract their coefficients ONLY. β’ Sometimes it may be necessary to simplify the radicals before you add or subtract. Simplify: β 99 + 6 11 β 2 3 β 9 β 11 + 6 11 β 2 3 β Simplify each radical separately, if necessary (as above) β 9 β 11 + 6 11 β2 3 β3 11 + 6 11 β 2 3 3 11 β 2 3 β Combine βlike radicalsβ Guided Practice: Multiplication of Radicals Tips: β’ This is NOT the only way to simplify these problems Simplify: 2 5 β 4 8 β Simplify each radical, if possible 2 5β8 2 2·8β 5β 2 β Group coefficients together, group radicands together β Multiply coefficients and radicands 16 10 β Simplify radical, if possible simplified Guided Practice: Division of Radicals Tips: β’ This is NOT the only way to simplify these problems Simplify: !"# ! ! !" β ! β Simplify each radical, if possible ! ! !" β ! ! ! ! ! ! ! ! ! β Divide numbers outside the radical ! 4 β Divide numbers under the radical no radical remaining β Simplify radical, if possible RADICALS IV.Simplify.Expressyouranswerinsimplifiedradicalform. 13)β 169 14) 80 β 14 5 15)β3 2 β 50 17) !" ! 16) !"# ! 18) 5 2 + 2 128 Geometric Notation and Definitions The following set of notation and definitions will be used throughout the entire course. Notation Meaning Diagram Line π΄π΅ or Line β β π΄π΅ or π΅π΄ or Line β π΄π΅ or π΅π΄ π΄π΅ Canβt switch order!! π΄π΅ or π΅π΄ = π΄ Has one dimension. Through any two points, there is exactly one line. Segment π΄π΅ π΄ Consists of two endpoints π΄ and π΅ and all of the points on π΄π΅ between π΄ and π΅ Ray π΄π΅ π΅ π΅ π΄ Consists of one endpoint π΄ and all the points on π΄π΅ that are on the same side as π΅ The length of segment π΄π΅ (has no segment bar on top) Equal to π΄ 5 π. π΅ πΆ 2 ππ. π· *πΏππππ‘βπ πππ πππππ ππππ π’πππ πππ πππ’ππ β π΄ *πππππππ‘π πππ ππππππ πππ ππππππ’πππ‘ Degree(s), a unit measure for angles Perpendicular β₯ Two lines that intersect to form a right angle. Parallel βπ΄π΅πΆ βπΆπ΅π΄ 2 ππ. πΉ Congruent !!!! πΆπ· β !!!! πΈπΉ 22° Congruent β‘π΄ β β‘π΅ ° β₯ πΈ π΅ 22° Equal πβ‘π΄ = πβ‘π΅ Angle π΄π΅πΆ has a vertex of B The vertex should be the middle letter πβ‘π΄π΅πΆ or The measure of angle π΄π΅πΆ πβ π΄π΅πΆ π΄π΅ = 5 π. Equal πΆπ· = πΈπΉ 2 = 2 Congruent (has the same measure) β‘π΄π΅πΆ or β π΄π΅πΆ π΅ π΄ π΅ πΆ π΄ πβ‘π΄π΅πΆ = 40° π΅ 40° πΆ 100° Two coplanar lines that never intersect. They have the same slope. Triangle π΄π΅πΆ π΅ π΄ πΆ Other Definitions Point A point has no dimension. Point π΄ It is represented by a dot. π΄ Plane A plane has two dimensions. It is represented by a shape that looks like a Plane π΄π΅πΆ floor or a wall and extends without end. Plane π π΄ Plane π΄π΅πΆorPlaneπ Sometimes, you can use a capital letter without a point, if it is provided. Two rays with a common end point that go in opposite directions. The first letter is the endpoint. Collinear Points Points that are on the same line. Coplanar Points Points that are on the same plane. Adjacent Angles Angles that share a common vertex and a common side that do not overlap each other. Angles must be coplanar. πΆ π΅ Through any three points not on the same line, there is exactly one plane. You can use three points, not all on the same line, to name it. Opposite Rays M π΄ π΄ π΅ πΆ !!!!!β andπ΅πΆ !!!!!β areoppositerays π΅π΄ See diagram above for OPPOSITE RAYS. π΄, π΅, πΆ are collinear points. See diagram for PLANE above. π΄, π΅, πΆ are coplanar points 3 1 2 β‘1andβ‘2areadjacent angles. β‘2andβ‘3areadjacent angles. β‘1andβ‘3areNOT adjacentangles. POINTS,LINES,&PLANES V.Determineifthestatementistrueorfalse. 19)C,D,Aarecollinearβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦...... True False 20)AandFarecollinearβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ True False 21)B,C,andFarecoplanarβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦. True False 22)IcanformexactlyoneplanewithpointsC,D,andEβ¦β¦β¦β¦β¦β¦True False 23)PlaneNandPlaneBCFintersectatlinelβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..True False 25)Giveanothernameforβ 24)GiveanothernameforπΆπΈ 26)GiveanothernameforπΆπΈ 27)GiveanothernameforPlaneBCF 28)GiventhatDisthemidpointofπΆπΈ,whatcanyou 29)Nameapairofoppositerays. conclude? Segments Congruent Segments - Line segments that have the same length. A βtick markβ is used on each segment to show that they are congruent. Segments are Lengths are equal B A congruent AB = CD !!!! β πΆπ· !!!! π΄π΅ D C Segment Addition Postulate - If point B is between A and C, then AB + BC = AC. Also, if AB + BC = AC, then point B is between point A and C. AC C B A 7 8 AB + BC = AC 7 + 8 = AC 15 = AC AB BC Midpoint - the point that divides a segment into two congruent segments. A M !!!! . Misthemidpointofπ΄π΅ !!!!!β ππ΅ !!!!!andAM=MB π΄π B Bisect - to cut a segment into 2 congruent segments Segment Bisector - a point, ray, line, line segment, or plane that intersects the segment as its midpoint. C β!!!!β !!!! πΆπ·isasegmentbisectorofπ΄π΅ M !!!!!β ππ΅ !!!!!andAM=MB π΄π B A D Perpendicular Bisector β a segment bisector that is perpendicular to the original segment. M A B SEGMENTS VI.Usethesegmentadditionpostulate. 30)Solveforx: 31)Solveforx,thenfindπΉπΊ. Identifythesegmentbisectorofπ·πΈthenfindPQ. 32) 33) Area and Perimeter Terms to know: β’ Area: how much is contained within a 2-dimensional, contained shape β’ Perimeter: the distance around a 2-dimensional, contained shape o for a circle, called circumference β’ Base and Height: one must be a side length. they must be perpendicular to each other β’ Area of a square (bβh), rectangle (bβh), triangle( ½bβh) , circle (Οr2) β’ Perimeter of polygons (sum of all the sides) β’ Circumference of a circle (2Οr) β’ Pythagorean Theorem: the relationship between the sides of a right triangle o a2 + b2 = c2, where c is the hypotenuse AreaandPerimeter VIII.FindtheareaANDperimeter/circumferenceofeachshape. 34) 35) 23m 16m 8in 8in 36) 37) 15cm 12cm 18mi