Prerequisite Skills Appendix

Transcription

1 Prerequisite Skills Appendi Adding Polnomials To add, add the like terms Add. a) b) c) 6 d) a a 8 a a 1 e) f) 6a b a b 7 Angle Properties To find the measure of, recall that the sum of the interior angles of a triangle is When a transversal intersects two parallel lines: 7 67 a) the alternate angles b) the corresponding angles are equal are equal c) the co-interior angles are supplementar + = 180 To find the measures of,, and z, use the fact that, since AC is parallel to EF, the alternate angles are equal. BEF ABE (alternate angles) BFE CBF (alternate angles) z 7 A E B 7 z F C 8 MHR Prerequisite Skills Appendi

2 z Find the unknown angle measures. a) b) a b c d 11 e g f c) d) w 6 w z 7 79 Common Factoring To factor the epression 10 8, determine the greatest common factor of both terms. Refer to greatest common factors in this appendi The greatest common factor is The second factor is or. The factors of 10 8 are and. Therefore, 10 8 ( ). 1. State the missing factor. a) 6 8 () b) () c) abc 10ab ab() d) 8a 1a a () e) ab 10c () f) 8 (). Factor. a) 1 b) 16 c) ab 6a d) 18 e) 6 f) 6 9 g) 8ab ab 1a b h) Prerequisite Skills Appendi MHR 9

3 Congruent Triangles Congruent triangles have the same shape and the same size. When two triangles are congruent, their corresponding angles and corresponding sides are equal. ABC is congruent to DEF. A D The following corresponding parts are equal. A D AB DE B E AC DF C F BC EF B C E ABC DEF F 1. List the corresponding equal parts in each pair of congruent scalene triangles. a) P S Q R T U b) A K B C M L Evaluating Epressions To evaluate the epression for and, substitute for and for in the epression. Then, simplif using the order of operations. () () () () Evaluate for,, and z 1. a) b) z c) ( z) d) z e) z f) g) (z ) h) z( ) i) z j) (z ) k) ( z) l) ( z) 60 MHR Prerequisite Skills Appendi

4 . Evaluate for,, and z 1. a) z b) c) z d) z 6 e) z f) z z g) z h) (z ) i) ( )( z) j) (z) k) z( ) l) z To complete the table of values for, substitute the given values for in and determine. 0 When, () () 10 1 When 0, (0) (0) When, () () Cop and complete each table of values. a) b) c) d) 1 e) f) Prerequisite Skills Appendi MHR 61

5 Evaluating Radicals Since 7 7 9, Since , Evaluate. a) b) c) d) 1.1 e) 0.09 f) g) h) 1.69 To evaluate 6, to the nearest tenth, use a calculator so 6 7., to the nearest tenth.. Evaluate, to the nearest tenth. a) b) 19 c) d) 0. e) 89. f) g) h) Epanding Epressions To epand ( ), use the distributive propert. ( ) ( ) 1 1. Epand. a) ( ) b) ( 7) c) (a b c) d) (a ) e) ( ) f) ( 6) g) ( 7) h) ( ) i) a(a a 1) Eponent Rules To multipl powers with the same base, add the eponents. To divide powers with the same base, subtract the eponents. To raise a power to a power, multipl the eponents. ( ) 8 6 MHR Prerequisite Skills Appendi

6 1. Simplif, using the eponent rules. Epress each answer in eponential form. a) b) 6 c) d) e) f) 7 g) 6 h) ( ) i) ( ) j) 7 k) z z l) m) z 8 z n) ( ) o) ( ) 8 p) q) ( )( ) r) 10m 7 (m ) s) ( ) t) ( ) First Differences First differences are calculated from tables of values in which the -coordinates are evenl spaced. First differences are found b subtracting consecutive -coordinates. If the first differences are constant, the relation is linear. If the first differences are not constant, the relation is non-linear. This relation is linear. First Differences 1 = 7 = = 9 This relation is non-linear. First Differences = 9 = = Use first differences to determine whether each relation is linear or non-linear. a) b) c) d) Prerequisite Skills Appendi MHR 6

7 Graphing Equations To graph the line 6 using a table of values, choose suitable values for. Complete a table of values b finding the value of for each value of Plot the points on a grid and draw a line through the points. 6 + = Graph each equation using a table of values. a) b) c) d) 1 To graph 6 using the intercepts, find the points where the graph of 6 crosses the - and -aes. To find the -intercept, let 0. 6 (0) 6 6 One point on the line is (6, 0). To find the -intercept, let Another point on the line is (0, ). Plot the points on a grid. Draw a line through the points. (0, ) + = 6 (6, 0) 0 6. Graph each equation using the intercepts. a) b) c) 8 d) 10 6 MHR Prerequisite Skills Appendi

8 To graph using the slope and -intercept, first find the point where the graph crosses the -ais. Since the -intercept is, plot the point (0, ). The slope is 1. Use the slope to find another point on the line. = + (1, ) (0, ) 0 Draw the graph.. Graph each equation using the slope and -intercept. a) b) c) d) 1 To find the point of intersection of the lines and, graph the lines on the same set of aes. The coordinates of the point of intersection are (, 1). (0, ) = (, 0) (, 0) 0 6 = (0, ). Graph each pair of lines and find the coordinates of the point of intersection. a) and 8 b) and 7 c) 6 and d) and Greatest Common Factors To determine the greatest common factor (GCF) of c d and 6cd, write each epression as a product. Then, write the factors that are common to both. c d c c d 6cd c d d d The GCF of c d and 6cd is c d or cd. 1. Determine the GCF of each pair. a), 10 b) 1, 8 c) 1z, 10z d) 0a, 0a e) 6, 1 f) 1ab, 7abc g) 1, 18 h) 9ab c, 7a bc. Determine the GCF of each set. a) ab, abc, 6a b) z, 9z, 1 c) 8, 0, d) 1mn, 18m n, 1mn e) 10rt, 1r t, 0r t f) 9rs, 18ab, 7 Prerequisite Skills Appendi MHR 6

9 Lengths of Line Segments The length of an line segment is a positive number. To find the length of the horizontal line segment joining (, ) and (, ), subtract the lesser -coordinate from the greater -coordinate. () 7 (, ) (, ) 0 The length of the horizontal line segment joining (, ) and (, ) is 7 units. 6 (, 7) To find the length of the vertical line segment joining (, ) and (, 7), subtract the -coordinates. 7 () 7 11 The length of the vertical line segment joining (, ) and (, 7) is 11 units. 0 (, ) 1. Find the length of the line segment joining each pair of points. a) (8, 7) and (, 7) b) (, 1) and (9, 1) c) (, 8) and (, ) d) (, 1) and (, 9) e) (, ) and (, ) f) (1, 7) and (1, 6) g) (, ) and (8, ) h) (, ) and (, 7) i) (6, ) and (0, ) j) (6, 7) and (6, 0) k) (6, ) and (8, ) l) (7, 9) and (7, ) Like Terms Like terms have eactl the same variables raised to eactl the same eponents. An epression is in simplest form when there are no like terms. To simplif 7, collect like terms Simplif b collecting like terms. a) 7 b) 6 8 c) d) a b c b a e) 7 f) t 8t 7 6t t g) 7 h) 11 7 i) 1 t t t t t 66 MHR Prerequisite Skills Appendi

10 Number Skills To evaluate epressions, use the order of operations (BEDMAS). ()(6) () 1 (9) 1 1 a 11 b 1 a11 b 1 a11 6 b 1 a b or 1 1. Evaluate each epression. a) () (6)(8) b) (1)(11) (9)(7) c) 10()() (1)(7)() d) e) 8 a7 f) 8 b a b a 8 b g) h).a 1 a b b.a b 1.a 7 b i) 0.(.) (6.7)(.1).(1.1). Order each set of fractions from least to greatest. a) b) 8,,, 7 1, 1, 7 8, 9, 6 7. Eplain wh the two epressions are not equal. a) b) c) 9 16, 9 16 ( ), 6, 7 9 Prerequisite Skills Appendi MHR 67

11 Percents The table shows how equivalent fractions, percents, and decimals can be epressed. Fraction Percent Decimal % = 8% = % = 10% Cop and complete the following table. Epress all fractions in lowest terms. a) b) Fraction Percent Decimal c) 8 d) e) f) g) h) i) % 0.0%.6% Polnomials The degree of a polnomial in one variable is the greatest power of the variable in an term. For the polnomial 7, the greatest power,, is contained in the term. 7 is a second-degree polnomial. 1. State the degree of each polnomial. a) b) c) d) 6m m e) 7 f) t t 9t 68 MHR Prerequisite Skills Appendi

12 Pthagorean Theorem To find the length of d in the right triangle, to the nearest tenth of a unit, use the Pthagorean theorem. This states that, in a right triangle, the square of the length of the hpotenuse is equal to the sum of the squares of the lengths of the other two sides. d 8 d 6 d 6 d 9 d 19 d The length of d is 6. units, to the nearest tenth of a unit. 1. In each right triangle, find the unknown side length, to the nearest tenth of a unit. a) b) c) 8 d 6 7 d) 6 e) f) Simplifing Epressions To simplif ( ) ( ), remove brackets and collect like terms. ( ) ( ) ( ) 1( ) Simplif. a) ( 7) b) (a 7) a c) ( ) d) ( ) 6 e) (t ) (t ) f) 8( ) ( 6) g) (z ) (z ) h) 7( w) (w ) i) 6( ) ( ) Prerequisite Skills Appendi MHR 69

13 . Simplif. a) ( ) ( ) b) (r s) (r s) c) (p q) (p q) d) ( ) (6 ) e) (a b) (7a b) f) (c d) c d g) a b c (a b c) h) ( z) 7 i) 6 ( z) Slope To find the slope of the line passing through the points (, 1) and (, ), use the slope formula. m () 6 (, ) rise 1 (, 1) run ( ) Find the slope of the line passing through each pair of points. a) (0, 0) and (, 6) b) (0, 0) and (, ) c) (1, ) and (, 6) d) (, ) and (, 7) e) (, 0) and (0, 6) f) (, ) and (, 6) g) (, ) and (, ) h) (, ) and (1, 8) i) (, 7) and (, 10) Parallel lines have the same slope. The slope of a line parallel to is. The product of the slopes of perpendicular lines is 1. 1 The slope of a line perpendicular to is, since State the slope of a line parallel to and a line perpendicular to each line. a) b) 1 c) 7 d) e) 1 f) 6 70 MHR Prerequisite Skills Appendi

14 Solving Equations To solve 11, isolate the variable. 11 Add to both sides: 11 Simplif: 1 1 Divide both sides b : Simplif: 7 To check, substitute 7 for in the original equation. L.S. R.S. 11 (7) 1 11 Since L.S. R.S., the solution is Solve and check. a) 11 b) 7 c) 17 d) 7 9 e) 1 1 f) g) h) 7 9 i) 9 To solve ( ) 9, epand to remove the brackets. ( ) 9 Epand: 6 9 Add 6 to both sides: Simplif: 1 Divide both sides b : Simplif: To check, substitute for in the original equation. L.S. ( ) R.S. 9 ( ) () 9 Since L.S. R.S., the solution is.. Solve and check. a) ( ) 10 b) ( 1) 16 c) 6( ) 7 1 d) ( 1) 1 e) ( ) ( 1) f) ( ) ( 7) g) 7( ) ( ) 0 h) ( ) ( 1) i) 7( 6) ( 7) 1 1 Prerequisite Skills Appendi MHR 71

15 1 To solve, eliminate fractions b multipling b the lowest common denominator, 1. 1 Multipl both sides b 1: Simplif: ( 1) ( ) Epand: 6 9 Add to both sides: 6 9 Simplif: 6 Subtract 6 from both sides: Simplif: Divide both sides b 1: To check, substitute for in the original equation. L.S. 1 R.S. 1 () 1 1 Since L.S. R.S., the solution is.. Solve and check. 1 a) 1 b) 6 1 c) d) e) f) g) 1 h) 0 1 i) 8 7 MHR Prerequisite Skills Appendi

16 Solving Proportions Proportions can be solved using the cross-product rule, which states a c that, if, then a d b c. b d To solve : :, first write the proportion in fraction form,. Then, use the cross-product rule. 1 1 or Solve for. Epress each answer as a fraction in lowest terms. a) b) 6 c) d) 7 e) : :6 f) 8: 6:7 g) :9 :6 h) :8 :. Solve for. Epress each answer as a decimal. Round to the nearest hundredth, if necessar..6 a). b).8..6 c). d) e) f) g) 6.: 7.: 8. h) 1.: 0..9: Prerequisite Skills Appendi MHR 7

17 Subtracting Polnomials To subtract 8 _ add the opposite of the polnomial that is being subtracted. The opposite of is Subtract. a) 6 8 b) c) 6 d) a a a 7a 8 e) a b 1 f) a b 6 _ 6 1 Transformations To translate ABC units to the left and units up, translate each verte of the triangle units to the left and units up. Join the new points to form the translation image, ABC. ABC and its translation image, ABC, are congruent, since corresponding side lengths and angles are equal. A' B' C' A B C 1. Draw each triangle on grid paper. Then, draw the translation image for the given translation. a) D b) R c) F P R E S Q T right, down left, down right, up 7 MHR Prerequisite Skills Appendi

18 To reflect DEF in the reflection line m, reflect each verte of DEF in the line. Each verte of DEF and its reflection image are the same perpendicular distance from the line m. Join the new points to form the reflection image, DEF. DEF and its reflection image, DEF, are congruent, since corresponding side lengths and angles are equal. E F D m D' F' E'. Draw each triangle on grid paper. Then, draw the reflection image in the given reflection line. a) b) m R T E D S C m c) P Q R m Prerequisite Skills Appendi MHR 7