Notes: Review of Algebra I skills


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1 Notes: Review of Algebra I skills Name: Date: Period:
2 Algebra Review: Systems of Equations
3 * If you divide or multiply an inequality by a negative number, then you must reverse the inequality sign!
4 Notes Factoring Trinomials: Process: 1. Draw a by box. a. Place the first term in the upper lefthand box. b. Place the third term in the lower righthand box.. Determine factors of the product a c that combine (add) to make b and place them in the opposite empty corners. (Use a factorsum table.) 3. Pull out the GCF from each row. (If b is negative, the larger GCF will be negative.) 4. Reduce the terms inside the boxes as you pull out the GCF. If the reduced terms in each column are the same, then you factored correctly! 5. The pulled out GCF s are one factor, and the reduced terms are the other factor. 6. To solve for x, set each factor = 0, and solve each equation for x. Example 1: Solve by factoring x 7x 18 = 0 x Factors Sum GCF: and reduce List factors Which pair of a c = 18. adds to b = 7? x x x x x x 18 1 and 18 1 and 18 and 9 and = = = = 7 9x x x 18 Example : Solve by factoring 3x 14x = 0 x 7x 18= 0 ( x 9)( x+ ) =0 x 9= 0 and x + = 0 x = 9 and x =
5 Simplifying Radicals When working with the simplification of radicals you must remember some basic information about perfect square numbers. You need to remember: Perfect Squares 4 = x 9 = 3 x 3 16 = 4 x 4 5 = 5 x 5 36 = 6 x 6 49 = 7 x 7 64 = 8 x 8 81 = 9 x = 10 x = 11 x = 1 x = 13 x = 14 x 14 5 = 15 x = 5 x 5 Radicals (square roots) = = 3 = 4 = 5 = 6 = 7 = 8 = 9 = = = = = 14 5 = = 5 While the list of squares and square roots goes on infinitely, the ones appearing in the charts above are the ones most commonly used. To simplify means to find another expression with the same value. It does not mean to find a decimal approximation.
6 To simplify (or reduce) a radical: 1. find the largest perfect square which will divide evenly into the number under your radical sign. This means that when you divide, you get no remainders, no decimals, no fractions. the largest perfect square that Reduce: 48 divides evenly into 48 is 16. If the number under your radical cannot be divided evenly by any of the perfect squares, your radical is already in simplest form and cannot be reduced further.. write the number appearing under your radical as the product (multiplication) of the perfect square and your answer from dividing. 48 = give each number in the product its own radical sign. 48 = 16 3 = reduce the "perfect" radical which you have now created. 48 = 16 3 = 16 3 = 4 5. you now have your answer = What happens if I do not choose the largest perfect square to start the process? If instead of choosing 16 as the largest perfect square to start this process, you choose 4, look what happens = = 4 1 = 4 1 = 1 Unfortunately, this answer is not in simplest form. The 1 can also be divided by a perfect square (4). 1 = 4 3 = 4 3 = 3 = 4 3 If you do not choose the largest perfect square to start the process, you will have to repeat the process. Example: Reduce: 3 50 Don't let the number in front of the radical distract you. It is simply "along for the ride" and will be multiplied times our final answer. The largest perfect square dividing evenly into 50 is = 3 5 = 3 5 Reduce the "perfect" radical and multiply times the 3 (who is "along for the ride") 3 5 = 3 5 = 15
7 Distance Formula, Midpoint & Slope Notes and Practice PreAP Geometry Name x = y = Give your answers in simplified radical form! Then also round if necessary. rise Δ y The triangle diagram can also help you find the midpoint and slope! Slope = =. run Δ x Slope of RS = Slope of ST = Midpoint = (average x, average y). To average numbers, ADD them and divide by. Midpoint of RS = Midpoint of ST =
8 Distance Formula, Midpoint & Slope Practice Name: Period: Show your work! For the distance, give your answer in simplified radical form, and also round to the nearest tenth. Draw and label a triangle Find the distance (length) Find the midpoint. Or, if given the Find the slope (write as for the given endpoints. between the endpoints. midpoint, find the missing endpoint. an improper fraction). 1. L(7, 0), Y(5, 9). B(4, 6), U(1, 3) 3. H(5, 7), K(9, 3) 4. W(1, 7), T(8, 4) 5. A(x 1, y 1 ), B(x, y ) Derive the distance formula. Derive the midpoint formula. Derive the slope formula. 6. D(?,?), F(5, 8). The midpoint is E(4, 3). Find D. 7. D(, 9), F(?,?). The midpoint is E(1, 6). Find F. 8. D(3, 8), F(?,?). The midpoint is E(1, ). Find F. 9. The coordinates of the vertices of a quadrilateral are R(1, 3), S(3, 3), T(5, 1), and U(, 1). Find the perimeter of the quadrilateral.
9 Algebra Practice with Segment Addition and Midpoint Geometry 15 Name: Date: Period: Segment Addition Postulate: If three points A, B, and C are collinear and B is between A and C, then Definition of midpoint: A midpoint is a point that divides a segment into. 1. Use the Segment Addition Postulate to write an equation and solve for x. If AB = 5, find the value of x. Then find AN and NB.. Use the definition of midpoint to write an equation and solve for x. M is the midpoint of RT. Find RM, MT, and RT. #36: Find the length of each segment. Tell whether the segments are congruent. 3. AC and BD 4. AD and BE 5. Find the midpoint of AD. 6. Find the midpoint of CD. 7. EG = 100. Find the value of x. Then find EF and FG. 8. Z is the midpoint of XY, and XY = 7. 4(x 5) (x + 15) Draw and label a picture, including congruency marks. Then find XZ. 9. If GJ = 3, find the value of x, GH and HJ. 10. If AX = 45, find the value of y, AQ, QX. 9x 8(x ) 11. Find PD if the coordinate of P is 7 and the coordinate of D is Find SK if the coordinate of S is 17 and the coordinate of K is Find the coordinate of B if AB = 8 and the coordinate of A is Find the coordinate of X if XY = 1 and the coordinate of Y is 0.
10 First, write 1 sentence explaining the relationship between the angles in each problem. (Hints: A linear pair adds to. Supplementary angles add to. Complementary angles add to. Right angles are. Vertical angles are. The sum of adjacent angles the whole. A bisector divides into two parts.) Next, set up an algebraic equation for this relationship. Finally, solve for x or the unknown angle. Show your work! 80 80
11 WS Drawing and Solving Angles and Segments PreAP Geometry sections 15, 16, and 17 Name: Date: Period: Draw and label a figure that represents each of these situations. Use congruence marks if possible. Then draw conclusions (if possible) about congruent angles or segments, angle measures, and segment lengths and are linear angles that are also and 4 are complementary but are not congruent. (include both of their measures) adjacent. m 3 = 50. (also include m 4 in your fig.) 19. a b and c b. (there will be 3 lines in this fig.) 0. 5 and 6 are supplementary and also a linear pair. 7 is vertical to 5. m 6 = 150. (include the measures of all 3 angles in your figure) 1. 8 and 9 are vertical angles. m 8 = is adjacent to 9 but m 10 = 0. (include the measures of all 3 angles in your figure). 11 and 1 are a linear pair. 13 is vertical to 11. m 1 = 100. (include all 3 angle measures) 3. d e but e f. (draw all 3 lines in your figure) Line f contains points G and H, and line e is the perpendicular bisector of GH and 15 are not adjacent but they are supplementary. 15 and 16 are adjacent and they are complementary. m 14 = 150. (include all 3 angle measures in your figure. 5. Ray GA bisects RGN. 6. B is the midpoint of AC. 7. Points R, S, and T are collinear (where S is between R and T). RS = 15 and ST = Points X, Y, and Z are collinear (where Y is between X and Z). YZ = 15 and XZ = 40.
12 9. Line p bisects CD at point E. 30. Line MK is the perpendicular bisector of ST. Find m 1 and m for each of the following conditions. Show your work. 31. m 1 is twice m and 3. m 1 is twice m and 1 and are complementary. 1 and are supplementary. 33. m 1 is three less than twice m and 1 and are complementary. 34. m 1 is three less than twice m and 1 and are supplementary. 35. m 1 is six more than five times m and 1 and are a linear pair. 36. m 1 is six more than five times m and 1 and are complementary. Draw a sketch and solve algebraically. Does every arrangement have a solution? 37. Given that AC = 4 and AB = (BC), 38. Given that AC = 18 and AB = 1 5 BC, find AB and BC if: find AB and BC if: a) B is between A and C. AB =, BC = a) B is between A and C. AB =, BC = b) C is between A and B. AB =, BC = b) C is between A and B. AB =, BC = c) A is between B and C. AB =, BC = c) A is between B and C. AB =, BC = 39. D lies in the interior of ABC; m ABC = 10, m ABD = x + 15, m DBC = 3x 5. Does BD bisect ABC? 40. MN bisects KMP; m KMN = 4x 3; m NMP = 3x + 9. Are KMN and NMP complementary angles?
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