GRADE 12 MATHEMATICS LEARNER NOTES

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1 SENIOR SECONDARY,059(0(7 PROGRAMME 0 GRADE LEARNER NOTES The SSIP is supported by c) Gauteng Department of Education, 0

2 TABLE OF CONTENTS LEARNER NOTES SESSION TOPIC Revision of Analytical Geometry (Grade ) PAGE c) Gauteng Department of Education, 0

3 GRADE SESSION 5 SESSION 5 TOPIC: REVISION OF ANALYTICAL GEOMETRY (GRADE ) Learner Note: Analytical Geometry is an important topic that carries a lot of marks in the matric final exam. Make sure that you know the basic formulae and then practise lots of examples involving applications of these formulae. The properties of quadrilaterals are extremely important in Analytical Geometry. Make sure you can prove that a quadrilateral is a parallelogram, rectangle, square, rhombus or trapezium by knowing the properties of these quadrilaterals. SECTION A: TYPICAL EXAM QUESTIONS QUESTION : 5 minutes In the diagram, PQRS is a trapezium with vertices P(5; ), Q(; ), R(9; 5) and S, and PS//QR. PT is the perpendicular height of PQRS and W is the midpoint of QR. Point S lies ˆ. on the x-axis and PRQ P(5;) S Q( ; ) T W R(9; 5) c) Gauteng Department of Education, 0

4 GRADE SESSION 5 QUESTION (a) (b) 5 4 ( ) E ; 7 E ; 7 E ; 7 x x y yc E ; A C ; A 7 xc yc E ; ; 7 xc yc or 7 xc or yc xc 6 or () 7 xc yc xc 6 yc 0 C(6 ; 0) yc 0 C(6 ; 0) (c) 4 0 ( ) mcd 6 5 mab mcd mab AB CD ( ) mbc 6 4 mad mbc mad mab mcd AB CD mad mbc AD BC ABCD is a parallelogram mab mad A 90 ABCD is a rectangle (0) AD BC ABCD is a parallelogram Now mab mad () ( ) AB AD A 90 ABCD is a rectangle (since one interior angle of parallelogram ABCD is 90 ) 4 c) Gauteng Department of Education, 0

5 GRADE SESSION 5 QUESTION (a) (b) ( ) k ; k ; ; k 6 k k 4 ; mab mcd lines // k ( ) k 5 5 (k ) 5 k 6 k k mab mcd lines (c) k 5 k 0 k 0 k mab mbc lines // (d) k ( ) k 6 6 ( k ) 6 k 8 k k 4 CD 5 And CD ( ) k (e) k k k k 0 k 6k 6 0 (k 8)(k ) k 8 or k ( ) k ; ; k k 4 k ( ) k k 5 k k ( ) k 4 CD ( ) k k k 0 k 6k 6 0 (k 8)(k ) k 8 or k [7] 5 c) Gauteng Department of Education, 0

6 (e) GRADE SESSION 5 QT ( ) ( ( )) QT 4 QT 5 QT 5 TR ( 9) ( ( 5)) TR 6 9 correct substitution to get QT answer for QT correct substitution to get TR answer for TR establishing that QT TR TR 45 TR TR 5 QT QT TR (f) tan tan 5 6, ˆ 7, TPR , 4 tan mpr ( 5) 5 9 tan 5 tan mpt tan 6, ˆ Now TPR ˆ TPR ˆ 5 6, TPR ˆ 7, TPR 90 7, , 4 [4] 6 c) Gauteng Department of Education, 0

7 (a) GRADE SESSION 5 (9) ( 5) W ; W(5 ; ) midpoint x5 () The equation of PW is x 5 (b) (c) 5 ( ) (PS QR) mps y ( x 5) 5 y x 9 y x mqr mpt mpt correct substitution into formula for equation y x 8 mqr (PT QR) y ( x 5) y x 0 y x 8 (d) mqr mps correct substitution into formula for equation 9 y x (4) correct substitution into formula for equation y x x x 8 x T(; ) y ( ) ( x ) y x y x x x 8 x 4 x 6 5 x 5 x y 8 T(; ) 7 c) Gauteng Department of Education, 0

8 GRADE SESSION 5 (a) (b) (c) (d) Determine the equation of PW if W is the midpoint of QR. Determine the equation of PS. Determine the equation of PT. Determine the coordinates of T. (e) Show that QT TR. (f) Calculate the size of rounded off to two decimal places. [4] QUESTION : () (4) 5 minutes Consider the following points on a Cartesian plane: A(;), B(;), C(-;k) and D(;-) Determine the value(s) of k if: (a) (b) (c) (d) (e) ( ; ) is the midpoint of AC. [7] AB is parallel to CD. AB CD. A, B and C are collinear. CD 5 QUESTION : 5 minutes ABCD is a quadrilateral with vertices A(;), B(;4), C and D(5; ). The diagonals BD and AC bisect each other at point E. B(;4) A(;) C D(5; ) 8 c) Gauteng Department of Education, 0

9 GRADE SESSION 5 SESSION 0 TOPIC: REVISION OF ANALYTICAL GEOMETRY (GRADE ) Learner Note: Analytical Geometry is an important topic that carries a lot of marks in the matric final exam. Make sure that you know the basic formulae and then practise lots of examples involving applications of these formulae. The properties of quadrilaterals are extremely important in Analytical Geometry. Make sure you can prove that a quadrilateral is a parallelogram, rectangle, square, rhombus or trapezium by knowing the properties of these quadrilaterals. SECTION A: TYPICAL EXAM QUESTIONS QUESTION : 5 minutes In the diagram, PQRS is a trapezium with vertices P(5; ), Q(; ), R(9; 5) and S, and PS//QR. PT is the perpendicular height of PQRS and W is the midpoint of QR. Point S lies ˆ. on the x-axis and PRQ P(5;) S Q( ; ) T W R(9; 5) 9 c) Gauteng Department of Education, 0

10 GRADE SESSION 5 Properties of Quadrilaterals Parallelogram: Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. Both pairs of opposite angles are equal. Both diagonals bisect each other. Rectangle: Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. Both pairs of opposite angles are equal. Both diagonals bisect each other. Diagonals are equal in length. All angles are right angles. Rhombus: Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. Both pairs of opposite angles are equal. Both diagonals bisect each other. All sides are equal in length. The diagonals bisect each other at 90. The diagonals bisect both pairs of opposite angles. 0 c) Gauteng Department of Education, 0

11 GRADE SESSION 5 SECTION B: ADDITIONAL CONTENT NOTES REVISION OF ANALYTICAL GEOMETRY (GRADE ) If AB is the line segment joining the points A( xa ; ya ) and B( xb ; yb ), then the following formulas apply to line segment AB. The Distance Formula AB ( xb xa ) ( yb ya ) or AB ( xb xa ) ( yb ya ) The Midpoint Formula x xb ya yb M A ; where M is the midpoint of AB. The Gradient of a line segment joining two points Gradient of AB yb ya xb xa Parallel lines Parallel lines have equal gradients. If AB CD then mab mcd Perpendicular lines The product of the gradients of two perpendicular lines is. If AB CD, then mab mcd The equation of the line Inclination of a line or y ya m x xa tan mab If mab 0, then is acute If mab 0, then is obtuse Collinear points (A, B and C) Using the gradient formula: mab mbc or mac mab or Using the distance formula: mac mbc c) Gauteng Department of Education, 0

12 (a) (b) (c) (d) (e) (b) SESSION 5 Determine the coordinates of E, the midpoint of BD. Determine the coordinates of C. Show that ABCD is a rectangle. Determine the area of ABCD. Calculate the size of the angle rounded off to the nearest degree. QUESTION 4: (a) GRADE 5 minutes Determine the numerical value of p if the straight line defined by the equation px y 6 has an angle of inclination of 5 with respect to the positive x-axis. Calculate the value of k if the points A(6;5), B(;) and C( k ; k 4) are collinear. QUESTION 5: () (0) (6) (8) [] (4) [7] 5 minutes In the diagram below, a(-4;5), C(-;-4) and B94;) are the vertices of a triangle in a Cartesian plane. CE AB with E on AB. E is the midpoint of line CD. AF BC with F on CB. The equation of AF is. (a) (b) (c) (d) Determine the equation of CD. Determine the coordinates of E. Determine the equation of the line through D and parallel to line AC. Determine, showing all calculations, whether the x-intercept of line CD also lies on the line through AF with the equation. (4) (6) [8] c) Gauteng Department of Education, 0

13 GRADE SESSION 5 QUESTION (a) (b) 5 4 ( ) E ; 7 E ; 7 E ; 7 x x y yc E ; A C ; A 7 xc yc E ; ; 7 xc yc or 7 xc or yc xc 6 or () 7 xc yc xc 6 yc 0 C(6 ; 0) yc 0 C(6 ; 0) (c) 4 0 ( ) mcd 6 5 mab mcd mab AB CD ( ) mbc 6 4 mad mbc mad mab mcd AB CD mad mbc AD BC ABCD is a parallelogram mab mad A 90 ABCD is a rectangle (0) AD BC ABCD is a parallelogram Now mab mad () ( ) AB AD A 90 ABCD is a rectangle (since one interior angle of parallelogram ABCD is 90 ) c) Gauteng Department of Education, 0

14 (d) GRADE SESSION 5 AB ( ) (4 ) AB ( ) (4 ) AB AB AD (5 ) ( ) AB AD Area ABCD (4 )( ) AB AD (5 ) ( ) Area ABCD 8 units AD 6 6 (6) AD 6 4 Area ABCD AD AB Area ABCD (4 )( ) 8 units B(;4) A(;) C(6;0) D(5; ) (e) tan mbd tan 4 ( ) 5 5 tan tan 5 tan 45 8, 4 ˆ 45 OCD (8) 4 c) Gauteng Department of Education, 0

15 GRADE SESSION 5 tan mdc 0 ( ) 6 5 tan 45 ˆ 45 OCD tan [] QUESTION 4 4(a) 4(b) y px 6 p y x p tan5 p p mab mbc p x p tan5 p p y (4) mab mbc working out gradients k 5 5 k 4 6 k k k k k k 5 [7] 5 c) Gauteng Department of Education, 0

16 GRADE SESSION 0 QUESTION 5 5(a) 5(b) Equation of AB Equation of CD correct substitution into formula for equation of line (4) E(;) 5(c) C(-;-4) E(;) D and correct substitution into formula for equation of line (6) Now Equation of line required: 5(d) Substitute (;0) LHS=RHS x-intercept: Substitute (;0) into lies on the line AF [8] 6 c) Gauteng Department of Education, 0

DISTANCE FORMULA: to find length or distance =( ) +( )

MATHEMATICS ANALYTICAL GEOMETRY DISTANCE FORMULA: to find length or distance =( ) +( ) A. TRIANGLES: Distance formula is used to show PERIMETER: sum of all the sides Scalene triangle: 3 unequal sides Isosceles