June 017 Dear Geometry Students and Parents: Welcome to Geometry! For the 017-018 school year, we would like to focus your attention to the fact that many concepts from Algebra I are infused into Geometry. In order to be successful in Geometry, a student must demonstrate a proficiency in certain skills including: Multiplying Polynomials Factoring Solving Quadratic Equations Solving Systems of Equations Simplifying Radicals Notation These topics will be reviewed briefly throughout the coming year; however, the topics are not re-taught in the Geometry course. To ensure that all students demonstrate the basic algebra skills to be successful, an assessment will be administered within the first two weeks of school (date to be determined). The attached review packet is provided for practice and is intended as a tool for assessment readiness. A one-day, full class period review will be conducted in each class. In addition, students are encouraged to seek extra help before or after school from their teacher for any topics requiring more personal in-depth remediation. Additional practice problems and explanations may also be found beginning on page 869 of the Geometry textbook. It is expected that each student will fully complete the review packet. Show all work on this packet. Packets will be collected in class on the day of the assessment. If you have any questions, please do not hesitate to contact your child s teacher. Ms. Alfaro Ms. Bravo Ms. Coons Ms. Dengel Ms. Fauerbach Ms. Mankikar Ms. Marchegiano Ms. McCorkell Mr. Peklo Ms. Perna Ms. Poyner Mr. Silverman Geometry Summer Review Packet Page 1
Binomial Products To multiply two binomials, you can use the Distributive Property systematically. You may know this as the FOIL method or Area Model. Example: Simplify ( )( ) ( ) ( ) ( ) ( ) Distribute Simplify--multiply Simplify--combine like terms Simplify ( ) ( )( ) Rewrite as the product of the binomial and itself ( ) ( ) ( )( ) ( )( ) Distribute Simplify--multiply Simplify--combine like terms Simplify 1) ( ) ) ( ) 3) ( ) Geometry Summer Review Packet Page
Factoring Factoring (Greatest Common Factor) Solution: The first step in ANY factoring problem is to factor out the GCF. The other factor is the original polynomial divided by the GCF. A variable must be in each term of the polynomial in order to be in the GCF. Always use the smallest exponent in the GCF. If there is no common factor, the polynomial can be prime. Example: Factor completely 4xy 18xy 3y 4xy 18xy 3y Initial polynomial GCF = 3 y Find the GCF 4xy 18xy 3y 3y 8x 6xy 1 Divide each term by the GCF 3y (8x 6xy 1) Final factorization Neither factor can be factored further (each factor is prime) Look to see if the polynomial factored further. Factor Completely 4) 3y 9y 5) 5 3 6 3 4 7 54x y z 9x y r 36x r 3 6) 5x y 9r 7z Geometry Summer Review Packet Page 3
Factoring (trinomial where a 1) Solution: Put expression into standard form, if not already done. Standard form: ax bx c ALWAYS start by attempting GCF facroting. If a is negative, factor out 1. Look for two factors of c that add up to b (i.e., factors f1 and f, such that f1 f c and f 1 f b ). The trinomial can then be factored as ( x f1)( x f ) When finding the factors f1 and f, remember that if c is negative, f1 and f must have different signs (one positive and one negative). If c is positive, then f1 and f must both have the same sign as b. If there are no two such factors, then the trinomial is prime. NOTE: if you factored out a GCF to start, then your original expression is NOT prime. Example: Factor completely x 0x 4 x 0x 4 Write in Standard Form, if necessary. ( x 10x 1) Factor out the GCF of. f 1 7, f 3 Find the factors of +1 that add up to -10. They must both be negative. ( x 7)( x 3) Use f1 and f to factor the trinomial. *As part of the final answer, don t forget to bring down the G.C.F.* Geometry Summer Review Packet Page 4
Factor Completely 7) 8) 9) 10) 11) 1) Geometry Summer Review Packet Page 5
Factoring (trinomial where a 1) Solution: There are several methods for factoring these types of trinomials. If you prefer another method, see your teacher if you have any questions. Example: Factor completely 5x 6 6x Factoring By Grouping 5x 6 6x Initial trinomial 6x 5x 6 Step 1: Put into standard form, if necessary not necessary Step : Factor out GCF or 1, if necessary 6 6 36 Step 3: Multiply a c f 4, 9 1 f Step 4: Find the factors of the product of a c that add up to b (i.e., factors f1 and f, such that f1 f = a c and f 1 f b ). Remember that if a c is negative, f1 and f must have different signs (one positive and one negative). If a c is positive, then f1 and f must both have the same sign as b. 6x 5x 6 6x 4x 9x 6 Step 5: Rewrite the middle term as the sum of two terms whose coefficients are the numbers above. 3x 33x x 3x x 3 ** Step 6: Factor by grouping **the expression in the parentheses must be the same Geometry Summer Review Packet Page 6
Factor completely using the method of your choice. 13) x 11x 5 14) 4x -14x 30 15) 8x 6x 9 16) -7x + 6x 17) 8x 6x 18) 1x 5 17x Geometry Summer Review Packet Page 7
Factoring (binomial) Difference of Two Perfect Squares Must be a binomial (two terms) Terms must be subtracted Each term must be a perfect square Solution: a b ( a b)( a b) **A binomial with the sum of two perfect squares is prime. Example: Factor completely 81x 16 81x 16 Original binomial not necessary Factor out G.C.F. or 1, if necessary 81x 16 Are the terms subtracted? YES! 81x (9x 16 (4) 81x 16 (9x 4)(9x 4) ) Are the first and last terms perfect squares? YES! Answer is the product of the sum and difference of the square roots. Factor completely 19) x 49 0) 5x 1 1) 100 y 36 ) 4 6 64y 49x Geometry Summer Review Packet Page 8
Solving Quadratics - There are several methods for solving quadratics. If you prefer another method, see your teacher if you have any questions. Solving Quadratics by Factoring Solution: Re-write in standard form ( ax bx c 0 ). Factor completely (use the methods above) Set each factor equal to zero (use or between the equations) Solve each resulting equation to obtain two solutions. Example: Solve 6x 5x 6 6x 5x 6 Original equation 6x 5x 6 0 Re-write in standard form ( x 3)(3x ) 0 Factor the polynomial completely (in this case, we used grouping) x 3 0 or 3x 0 Set each factor equal to zero 3 x or x 3 Solve both equations Tip: Make a positive first, and don t forget to start factoring by looking for a GCF! Solving Quadratics by Quadratic Formula x = Solution: The quadratic formula is generally used to solve all quadratics in the form and will find the solution to the problem every time. However, there are times when other solving techniques are faster and should be considered. Geometry Summer Review Packet Page 9
Solve by the method of your choice. 3) 16 x 10x 4) x 70 4x 5) 4x 10x 6 0 6) 7) 8) 9) 30) 31) 3) Geometry Summer Review Packet Page 10
System of Equations There are three ways to solve a system of equations: 1. Graphing Method (useful when equations are in slope-intercept form). Substitution Method (useful when at least variable has a coefficient of 1) 3. Elimination Method (useful when equations don t fit any of the above forms) We will demonstrate the substitution and elimination methods. Example: Consider the system of equations: { Solution: Substitution method: Step 1: Solve for one variable in one of the equations. Step : Substitute the expression into the OTHER equation. Step 3: Solve your new equation. Step 4: Substitute the value of variable into one of the original equations to determine second variable. Step 5: Write your answer as an ordered pair. Step 1: Solve for one variable ( ) Substitute Substitute Solution: ( ) Geometry Summer Review Packet Page 11
Elimination Method: Step 1: In both equations, line up the variables on one side of the equals sign and the constants on the other side. Step : Determine which variable you want to eliminate you will be working to get the coefficients to be the same number with opposite signs. Step : As necessary, multiply one or both equations by a number to make the coefficients to be the same number with opposite signs. Step 3: Add the equations together so that one variable cancels out. Step 4: Solve the equation resulting equation in one variable. Step 5: Substitute the value of variable into one of the original equations to determine the value of the second variable Step 5: Write your answer as an ordered pair. { Eliminate the x variable by multiplying the second equation by ( ) ( ) Substitute Solution: ( ) Geometry Summer Review Packet Page 1
Solve the following system of equations by substitution. 33) { 34) { Solve the following system of equations by elimination. 35) { 36) { Solve the following system of equations by elimination or substitution. 37) { 38) { Geometry Summer Review Packet Page 13
Radicals Simplify Radicals Radical symbol Solutions: You will need to know all your perfect squares through. To simplify - - square root of perfect square - Examples: Simplify the following 1. ) 3) =non-real answer 4) = = Geometry Summer Review Packet Page 14
Add/Subtract Radicals Solution: In order to add or subtract radicals, the index and the radicand must be the same! You combine radicals as if you were combining like-terms add/subtract their coefficients ONLY. Sometimes it may be necessary to simplify the radicals before you add or subtract. Examples: Simplify. Express your answer in simplified radical form. 1) ) 3) Geometry Summer Review Packet Page 15
Multiplication/Division of Radicals Solution: **This is NOT the only way to simplify these problems** Simplify the radical first. For multiplication: Multiply numbers outside the radical together Multiply numbers under the radical together Make sure to check if your radical needs to be simplified again For division: Divide numbers outside the radical Divide numbers under the radical Make sure to check if your radical needs to be simplified again Examples: Simplify. Express your answer in simplified radical form. 1) ) 3) ( ) ( )( )( ) Don t forget to simplify the radical Geometry Summer Review Packet Page 16
Simplify. Express your answer in simplified radical form. 39) 40) 41) 4) 43) 44) 45) 46) 47) 48) ( ) Geometry Summer Review Packet Page 17
Notation and Definitions The following set of notation and definitions will be used throughout the entire course. Notation Meaning Diagram or Line or Line or Line or Can t switch order!! Or Has one dimension. Through any two points, there is exactly one line. Segment Consists of two endpoints and and all of the points on between and Ray Consists of one endpoint and all the points on that are on the same side as The length of segment (has no segment bar on top) Equal to * A A A A B B m. B AB m. C in. D E in. F Equal CD EF B Congruent CD EF Congruent (has the same measure) * A B or or Angle The measure of angle Equal m A m B A B C A Congruent A B Degree(s), a unit measure for angles Perpendicular B C m ABC Two lines that intersect to form a right angle. Parallel Two lines that never intersect. They have the same slope. Triangle B A C Geometry Summer Review Packet Page 18
Other Definitions Point Point A point has no dimension. It is represented by a dot. A Plane Plane A plane has two dimensions. It is represented by a shape that looks like a floor or a wall and extends without end. Plane 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 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. A B C BA and BC are opposite rays Collinear Points Points that are on the same line. See diagram above for OPPOSITE RAYS. are collinear points. Coplanar Points Points that are on the same plane. See diagram for PLANE above. are coplanar points Adjacent Angles Angles that share a common vertex and a common side and angles. are adjacent and are adjacent angles. and are NOT adjacent angles. Geometry Summer Review Packet Page 19
Algebra 1 Review Packet Answer Sheet 1) ) 3) 4) ( ) 5) ( ) 6) prime 7) ( )( ) 8) ( )( ) 9) ( )( ) 10) prime 11) ( )( ) 1) ( )( ) 13) ( )( ) 14) (x + 3)(x -5) 15) ( )( ) 16) (3x -)(x 1) 17) ( )( ) 18) (4x +1)(3x - 5) 19) (x -7)(x +7) 0) ( )( ) 1) ( )( ) or ( )( ) ) prime 3) x = -8 or x = - 4) x = -7 or x = -5 5) x = or x = 6) r = 6 or r = 5 7) n = or n = 8) n = 4 or n = -1 9) r = 1 Geometry Summer Review Packet Page 0
30)n = or n = 31) k = 0 or k = 3) a = - or a = 33) (, 5) 34) (4, -1) 35) or No Solution 36) ( ) 37) infinite solutions 38) (1,) 39) 40) 41) -30 4) 43) 44) 45) 46) 8-47) 48) 147 Geometry Summer Review Packet Page 1