GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 11 SESSION 19 LEARNER NOTES

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Andra Crawford
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1 ANALYTIAL GEOMETRY () Learner Note: Analytical Geometry is an imortant toic that carries a lot of marks in the matric final exam. Make sure that you know the basic formulae and then ractise lots of examles involving alications of these formulae. The roerties of quadrilaterals are extremely imortant in Analytical Geometry. Make sure you know how to rove that a quadrilateral is a arallelogram, rectangle, square, rhombus or traezium by knowing the roerties of these quadrilaterals. SETION A: TYPIAL EXAM QUESTIONS Question 1 The straight line through PQ with equation x y 1 cuts the y-axis at Q. The straight line through PR with equation xy 1 cuts the x-axis at R. The coordinates of P are (;). P(;) Determine: (a) the coordinates of Q and R. () (b) the equation of QR. () (c) the coordinates of oint S, the midoint of QR. () (d) the equation of PS. (1) (e) the size of angle rounded off to the nearest degree. (Use the diagram for indicating angles) (6) (f) the area of PQR rounded off to two decimal laces. (6) Page 1 of 8

2 SETION B: SOLUTIONS AND HINTS (a) oordinates of Q: (0) y 1 y 1 Q(0; 1) oordinates of R: x (0) 1 x 1 x 4 R(4;0) (b) Q(0; 1) R(4;0) 0 ( 1) 1 mqr y x1 4 (c) S ; 1 S ; Q(0; 1) R(4;0) m QR y-intercet: 1 1 y x 1 4 formula correct coordinates () () () (d) Equation of PS: x answer (1) (e) tan m PQ ( 1) tan 0 tan 6 or tan m PQ Now x y 1 y x1 y x1 tan 6 Using formula for inclination tan 6 tan (6) Page of 8

3 tan m PR 0 tan 4 tan or tan m PR Now xy 1 y x1 y x 6 tan (f) PQ ( 0) ( ( 1)) PQ 4 16 PQ 0 PQ 0 PR ( 4) ( 0) PR 4 9 PR 1 PR 1 1 Area PQR 0 1 sin 61 Area PQR 7, 05 units correct substitution for PQ PQ 0 correct substitution for PR PR 1 Area rule 7,05 (6) Page of 8

4 SETION : ADDITIONAL ONTENT NOTES If is the line segment joining the oints A( xa; ya) and B( xb; y B), then the following formulas aly to line segment. The Distance Formula xb x A yb ya ( ) ( ) or ( x B xa ) ( yb ya) The Midoint Formula x x y y M ; A B A B where M is the midoint of. The Gradient of a line segment joining two oints B A Gradient of y y x x B A Parallel lines Parallel lines have equal gradients. If D then m md Perendicular lines The roduct of the gradients of two erendicular lines is 1. If D, then m md 1 The equation of the line y y m x x A A Inclination of a line tan m If m 0, then is acute If m 0, then is obtuse ollinear oints (A, B and ) m mb or m m or A m A m B Page 4 of 8

5 SETION D: HOMEWORK Question 1 In the diagram below, A( 4 ;5), ( 1; 4) and B(4 ;1) are the vertices of a triangle in a artesianlane. E with E on. E is the midoint of line D. AF B with F on B. The equation of AFis x y 1. D A( 4 ; 5) E B(4;1) F (a) Determine the equation of D. (4) (b) Determine the coordinates of E. (5) (c) Determine the equation of the line through D arallel to line A. (6) (d) Determine, showing all calculations, whether the x-intercet of line D also lies on the line through AF with equation x y 1. () Question onsider the following oints on a artesian lane: A(1; ), B(;1), ( ; k) and D( ; ). Determine the value(s) of k if: ( 1 ; 4) (a) ( 1; ) is the midoint of A. () (b) is arallel to D. () (c) D. () (d) A, B and are collinear. () (e) D 5 (5) Page 5 of 8

6 SETION E: SOLUTIONS TO SESSION 18 HOMEWORK (a) 5 4 ( 1) E ; 7 E ; 7 E ; () (b) 7 xa x ya y E ; ; 7 1x y E ; ; 7 1x y or 7 1 x or y x 6 or y 0 (6 ; 0) 7 1 x y x 6 y 0 (6 ; 0) (5) (c) 4 1 m ( 1) 1 md m m D AD D ( 1) 4 mad mb m m B AD B D is a arallelogram Now m m (1) ( 1) 1 AD AD Â 90 D is a rectangle (since one interior angle of arallelogram D is 90 ) m m D D m AD m B AD B D is a arallelogram m mad 1 Â 90 D is a rectangle (10) Page 6 of 8

7 (d) ( 1) (4 ) 11 AD (5 1) ( 1 ) AD AD 16 4 Area D AD Area D (4 )( ) 8 units ( 1) (4 ) AD (5 1) ( 1 ) AD Area D (4 )( ) Area D 8 units (6) (e) tan m BD 4 ( 1) tan 5 5 tan tan m D 0 ( 1) 1 tan tan 1 45 OD ˆ tan 11 tan ,4 ˆ OD (8) Page 7 of 8

8 B(;4) A(1;) (6;0) D(5; 1) (a) y x 6 y x tan15 1 m m (b) B 5 k 4 6 k k 1 k k k k 5 y x tan15 1 m mb working out gradients k 5 (4) () The SSIP is suorted by Page 8 of 8

CN#4 Biconditional Statements and Definitions

CN#4 Biconditional Statements and Definitions CN#4 s and Definitions OBJECTIVES: STUDENTS WILL BE ABLE TO WRITE AND ANALYZE BICONDITIONAL STATEMENTS. Vocabulary biconditional statement definition polygon triangle quadrilateral When you combine a conditional