Page 1 Co-ordinate Geometry 1) Find the distance between two points. a) (4, 7) and (-3,2) b) (0,6) and (8,0) Solution: a) (4, 7) and (-3, 2) Let A(x 1 ,y 1 ) ? (4,7) and B(x 2 ,y 2 ) ?(-3,2) Distance between the points A(4,7) and B(-3,2) is given by AB = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 2 2 7 2 4 ) 3 ( ? ? ? ? = ? ? ? ? 2 2 5 7 ? ? ? = 49 25 ? = 74 Distance between the points A(4,7) and B(-3,2) is 74 units. b) (0,6) and (8,0) P(x 1 ,y 1 ) ? (0,6) Q(x 2 ,y 2 ) ?(8,0) Distance between the points P(0,6) and Q(8,0) is given by PQ = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 (8 0 0 6 ? ? ? = ? ? ? ? 22 86 ?? = 64 36 ? = 100 = 10 Distance between the points A(0,6) and Q(8,0) is 10 units. 2) Determine if the points (3,-1), (0,5) and (-2,-8) are collinear. Solution: Consider, D(x 1 ,y 1 ) = (3,-1), E(x 2 ,y 2 ) = (0,5) & F(x 3 ,y 3 )=(-2,-8). Distance between the points D (3,-1) and E (0,5) is given by Page 2 Co-ordinate Geometry 1) Find the distance between two points. a) (4, 7) and (-3,2) b) (0,6) and (8,0) Solution: a) (4, 7) and (-3, 2) Let A(x 1 ,y 1 ) ? (4,7) and B(x 2 ,y 2 ) ?(-3,2) Distance between the points A(4,7) and B(-3,2) is given by AB = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 2 2 7 2 4 ) 3 ( ? ? ? ? = ? ? ? ? 2 2 5 7 ? ? ? = 49 25 ? = 74 Distance between the points A(4,7) and B(-3,2) is 74 units. b) (0,6) and (8,0) P(x 1 ,y 1 ) ? (0,6) Q(x 2 ,y 2 ) ?(8,0) Distance between the points P(0,6) and Q(8,0) is given by PQ = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 (8 0 0 6 ? ? ? = ? ? ? ? 22 86 ?? = 64 36 ? = 100 = 10 Distance between the points A(0,6) and Q(8,0) is 10 units. 2) Determine if the points (3,-1), (0,5) and (-2,-8) are collinear. Solution: Consider, D(x 1 ,y 1 ) = (3,-1), E(x 2 ,y 2 ) = (0,5) & F(x 3 ,y 3 )=(-2,-8). Distance between the points D (3,-1) and E (0,5) is given by Co-ordinate Geometry DE = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 (0 3 5 ( 1) ? ? ? ? = ? ? ? ? 22 36 ?? = 9 36 ? DE = 45 Distance between the points E(0,5) and F(-2,-8) is given by EF = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? = ? ? ? ? 22 ( 2) 0 ( 8) 5 ? ? ? ? ? = ? ? ? ? 22 2 13 ? ? ? = 4 169 ? EF = 173 Distance between the points D(3,-1) and F(-2,-8) is given by DF = ? ? ? ? 22 3 1 3 1 x x y y ? ? ? = ? ? ? ? 22 ( 2) 3 ( 8) ( 1) ? ? ? ? ? ? = ? ? ? ? 22 57 ? ? ? = 25 49 ? DF = 74 DE+EF= 45 74 ? ? 173 =DF Hence D, E and F are not collinear. 3) Check whether (5,4) (6,9) and (11,10) form the vertices of an isosceles triangle. Solution: Consider, A(x 1 ,y 1 ) = (5,4), B(x 2 ,y 2 ) = (6,9) C(x 3 ,y 3 )=(11,10) Distance between the points A(5,4) and B(6,9) is Page 3 Co-ordinate Geometry 1) Find the distance between two points. a) (4, 7) and (-3,2) b) (0,6) and (8,0) Solution: a) (4, 7) and (-3, 2) Let A(x 1 ,y 1 ) ? (4,7) and B(x 2 ,y 2 ) ?(-3,2) Distance between the points A(4,7) and B(-3,2) is given by AB = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 2 2 7 2 4 ) 3 ( ? ? ? ? = ? ? ? ? 2 2 5 7 ? ? ? = 49 25 ? = 74 Distance between the points A(4,7) and B(-3,2) is 74 units. b) (0,6) and (8,0) P(x 1 ,y 1 ) ? (0,6) Q(x 2 ,y 2 ) ?(8,0) Distance between the points P(0,6) and Q(8,0) is given by PQ = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 (8 0 0 6 ? ? ? = ? ? ? ? 22 86 ?? = 64 36 ? = 100 = 10 Distance between the points A(0,6) and Q(8,0) is 10 units. 2) Determine if the points (3,-1), (0,5) and (-2,-8) are collinear. Solution: Consider, D(x 1 ,y 1 ) = (3,-1), E(x 2 ,y 2 ) = (0,5) & F(x 3 ,y 3 )=(-2,-8). Distance between the points D (3,-1) and E (0,5) is given by Co-ordinate Geometry DE = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 (0 3 5 ( 1) ? ? ? ? = ? ? ? ? 22 36 ?? = 9 36 ? DE = 45 Distance between the points E(0,5) and F(-2,-8) is given by EF = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? = ? ? ? ? 22 ( 2) 0 ( 8) 5 ? ? ? ? ? = ? ? ? ? 22 2 13 ? ? ? = 4 169 ? EF = 173 Distance between the points D(3,-1) and F(-2,-8) is given by DF = ? ? ? ? 22 3 1 3 1 x x y y ? ? ? = ? ? ? ? 22 ( 2) 3 ( 8) ( 1) ? ? ? ? ? ? = ? ? ? ? 22 57 ? ? ? = 25 49 ? DF = 74 DE+EF= 45 74 ? ? 173 =DF Hence D, E and F are not collinear. 3) Check whether (5,4) (6,9) and (11,10) form the vertices of an isosceles triangle. Solution: Consider, A(x 1 ,y 1 ) = (5,4), B(x 2 ,y 2 ) = (6,9) C(x 3 ,y 3 )=(11,10) Distance between the points A(5,4) and B(6,9) is Co-ordinate Geometry AB = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 6 5 9 4 ? ? ? = ? ? ? ? 22 15 ? = 1 25 ? AB = 26 Distance between the points B (6,9) and C (11,10) is BC = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? = ? ? ? ? 22 11 6 10 9 ? ? ? = ? ? ? ? 22 51 ? = 25 1 ? = 26 Therefore, AB=BC Hence A (5,4), B (6,9) and C (11,10) form the vertices of an isoscelestriangle. 4) What type of quadrilaterals to the points (-2,-1) (2,0) (-2,1) and (-3,0) form. Solution: Let P(x 1 ,y 1 ) = (-2, -1), Q(x 2 ,y 2 ) = (2,0), R(x 3 ,y 3 ) = (-2,1) &S(x 4 ,y 4 ) = (-3,0). Distance between the points P(-2,-1) and Q(2,0) is PQ = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 2 ( 2) 0 ( 1) ? ? ? ? ? = ? ? ? ? 22 41 ? = 16 1 ? PQ = 17 Distance between the points Q(2,0) and R(-2,1) is QR = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? Page 4 Co-ordinate Geometry 1) Find the distance between two points. a) (4, 7) and (-3,2) b) (0,6) and (8,0) Solution: a) (4, 7) and (-3, 2) Let A(x 1 ,y 1 ) ? (4,7) and B(x 2 ,y 2 ) ?(-3,2) Distance between the points A(4,7) and B(-3,2) is given by AB = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 2 2 7 2 4 ) 3 ( ? ? ? ? = ? ? ? ? 2 2 5 7 ? ? ? = 49 25 ? = 74 Distance between the points A(4,7) and B(-3,2) is 74 units. b) (0,6) and (8,0) P(x 1 ,y 1 ) ? (0,6) Q(x 2 ,y 2 ) ?(8,0) Distance between the points P(0,6) and Q(8,0) is given by PQ = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 (8 0 0 6 ? ? ? = ? ? ? ? 22 86 ?? = 64 36 ? = 100 = 10 Distance between the points A(0,6) and Q(8,0) is 10 units. 2) Determine if the points (3,-1), (0,5) and (-2,-8) are collinear. Solution: Consider, D(x 1 ,y 1 ) = (3,-1), E(x 2 ,y 2 ) = (0,5) & F(x 3 ,y 3 )=(-2,-8). Distance between the points D (3,-1) and E (0,5) is given by Co-ordinate Geometry DE = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 (0 3 5 ( 1) ? ? ? ? = ? ? ? ? 22 36 ?? = 9 36 ? DE = 45 Distance between the points E(0,5) and F(-2,-8) is given by EF = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? = ? ? ? ? 22 ( 2) 0 ( 8) 5 ? ? ? ? ? = ? ? ? ? 22 2 13 ? ? ? = 4 169 ? EF = 173 Distance between the points D(3,-1) and F(-2,-8) is given by DF = ? ? ? ? 22 3 1 3 1 x x y y ? ? ? = ? ? ? ? 22 ( 2) 3 ( 8) ( 1) ? ? ? ? ? ? = ? ? ? ? 22 57 ? ? ? = 25 49 ? DF = 74 DE+EF= 45 74 ? ? 173 =DF Hence D, E and F are not collinear. 3) Check whether (5,4) (6,9) and (11,10) form the vertices of an isosceles triangle. Solution: Consider, A(x 1 ,y 1 ) = (5,4), B(x 2 ,y 2 ) = (6,9) C(x 3 ,y 3 )=(11,10) Distance between the points A(5,4) and B(6,9) is Co-ordinate Geometry AB = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 6 5 9 4 ? ? ? = ? ? ? ? 22 15 ? = 1 25 ? AB = 26 Distance between the points B (6,9) and C (11,10) is BC = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? = ? ? ? ? 22 11 6 10 9 ? ? ? = ? ? ? ? 22 51 ? = 25 1 ? = 26 Therefore, AB=BC Hence A (5,4), B (6,9) and C (11,10) form the vertices of an isoscelestriangle. 4) What type of quadrilaterals to the points (-2,-1) (2,0) (-2,1) and (-3,0) form. Solution: Let P(x 1 ,y 1 ) = (-2, -1), Q(x 2 ,y 2 ) = (2,0), R(x 3 ,y 3 ) = (-2,1) &S(x 4 ,y 4 ) = (-3,0). Distance between the points P(-2,-1) and Q(2,0) is PQ = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 2 ( 2) 0 ( 1) ? ? ? ? ? = ? ? ? ? 22 41 ? = 16 1 ? PQ = 17 Distance between the points Q(2,0) and R(-2,1) is QR = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? Co-ordinate Geometry = ? ? ? ? 22 ( 2) 2 1 0 ? ? ? ? = ? ? ? ? 22 41 ?? = 16 1 ? QR = 17 Distance between the points R (-2,1) and S (-3,0) is RS = ? ? ? ? 22 4 3 4 3 x x y y ? ? ? = ? ? ? ? 22 ( 3) ( 2) 0 1 ? ? ? ? ? = ? ? ? ? 22 11 ? ? ? = 11 ? RS = 2 Distance between the points S(-3,0) and P(-2,-1) is SP = ? ? ? ? 22 4 1 4 1 x x y y ? ? ? = ? ? ? ? 22 ( 2) ( 3) ( 1) 0 ? ? ? ? ? ? = ? ? ? ? 22 11 ?? = 11 ? SP = 2 Therefore PQ=QR and RS=SP Thus Opposites sides of the quadrilateral PQRS are equal. Therefore by definition, PQRS is a parallelogram 5) Show that the points (1,7) (4,2) (-1,-1) and (-4,4) form the vertices of a square. Solution: Consider, A(x 1 ,y 1 ) ? (1,7), B(x 2 ,y 2 ) ?(4,2), C(x 3 ,y 3 ) ? (-1, -1)& D(x 4 ,y 4 ) ? (-4,4) Distance between the points A(1,7) and B(4,2) is AB = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 4 1 2 7 ? ? ? Page 5 Co-ordinate Geometry 1) Find the distance between two points. a) (4, 7) and (-3,2) b) (0,6) and (8,0) Solution: a) (4, 7) and (-3, 2) Let A(x 1 ,y 1 ) ? (4,7) and B(x 2 ,y 2 ) ?(-3,2) Distance between the points A(4,7) and B(-3,2) is given by AB = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 2 2 7 2 4 ) 3 ( ? ? ? ? = ? ? ? ? 2 2 5 7 ? ? ? = 49 25 ? = 74 Distance between the points A(4,7) and B(-3,2) is 74 units. b) (0,6) and (8,0) P(x 1 ,y 1 ) ? (0,6) Q(x 2 ,y 2 ) ?(8,0) Distance between the points P(0,6) and Q(8,0) is given by PQ = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 (8 0 0 6 ? ? ? = ? ? ? ? 22 86 ?? = 64 36 ? = 100 = 10 Distance between the points A(0,6) and Q(8,0) is 10 units. 2) Determine if the points (3,-1), (0,5) and (-2,-8) are collinear. Solution: Consider, D(x 1 ,y 1 ) = (3,-1), E(x 2 ,y 2 ) = (0,5) & F(x 3 ,y 3 )=(-2,-8). Distance between the points D (3,-1) and E (0,5) is given by Co-ordinate Geometry DE = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 (0 3 5 ( 1) ? ? ? ? = ? ? ? ? 22 36 ?? = 9 36 ? DE = 45 Distance between the points E(0,5) and F(-2,-8) is given by EF = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? = ? ? ? ? 22 ( 2) 0 ( 8) 5 ? ? ? ? ? = ? ? ? ? 22 2 13 ? ? ? = 4 169 ? EF = 173 Distance between the points D(3,-1) and F(-2,-8) is given by DF = ? ? ? ? 22 3 1 3 1 x x y y ? ? ? = ? ? ? ? 22 ( 2) 3 ( 8) ( 1) ? ? ? ? ? ? = ? ? ? ? 22 57 ? ? ? = 25 49 ? DF = 74 DE+EF= 45 74 ? ? 173 =DF Hence D, E and F are not collinear. 3) Check whether (5,4) (6,9) and (11,10) form the vertices of an isosceles triangle. Solution: Consider, A(x 1 ,y 1 ) = (5,4), B(x 2 ,y 2 ) = (6,9) C(x 3 ,y 3 )=(11,10) Distance between the points A(5,4) and B(6,9) is Co-ordinate Geometry AB = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 6 5 9 4 ? ? ? = ? ? ? ? 22 15 ? = 1 25 ? AB = 26 Distance between the points B (6,9) and C (11,10) is BC = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? = ? ? ? ? 22 11 6 10 9 ? ? ? = ? ? ? ? 22 51 ? = 25 1 ? = 26 Therefore, AB=BC Hence A (5,4), B (6,9) and C (11,10) form the vertices of an isoscelestriangle. 4) What type of quadrilaterals to the points (-2,-1) (2,0) (-2,1) and (-3,0) form. Solution: Let P(x 1 ,y 1 ) = (-2, -1), Q(x 2 ,y 2 ) = (2,0), R(x 3 ,y 3 ) = (-2,1) &S(x 4 ,y 4 ) = (-3,0). Distance between the points P(-2,-1) and Q(2,0) is PQ = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 2 ( 2) 0 ( 1) ? ? ? ? ? = ? ? ? ? 22 41 ? = 16 1 ? PQ = 17 Distance between the points Q(2,0) and R(-2,1) is QR = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? Co-ordinate Geometry = ? ? ? ? 22 ( 2) 2 1 0 ? ? ? ? = ? ? ? ? 22 41 ?? = 16 1 ? QR = 17 Distance between the points R (-2,1) and S (-3,0) is RS = ? ? ? ? 22 4 3 4 3 x x y y ? ? ? = ? ? ? ? 22 ( 3) ( 2) 0 1 ? ? ? ? ? = ? ? ? ? 22 11 ? ? ? = 11 ? RS = 2 Distance between the points S(-3,0) and P(-2,-1) is SP = ? ? ? ? 22 4 1 4 1 x x y y ? ? ? = ? ? ? ? 22 ( 2) ( 3) ( 1) 0 ? ? ? ? ? ? = ? ? ? ? 22 11 ?? = 11 ? SP = 2 Therefore PQ=QR and RS=SP Thus Opposites sides of the quadrilateral PQRS are equal. Therefore by definition, PQRS is a parallelogram 5) Show that the points (1,7) (4,2) (-1,-1) and (-4,4) form the vertices of a square. Solution: Consider, A(x 1 ,y 1 ) ? (1,7), B(x 2 ,y 2 ) ?(4,2), C(x 3 ,y 3 ) ? (-1, -1)& D(x 4 ,y 4 ) ? (-4,4) Distance between the points A(1,7) and B(4,2) is AB = ? ? ? ? 22 2 1 2 1 x x y y ? ? ? = ? ? ? ? 22 4 1 2 7 ? ? ? Co-ordinate Geometry = ? ? ? ? 22 35 ?? = 9 25 ? AB = 34 units.................... (1) Distance between the points B(4,2) and C(-1,-1) is BC = ? ? ? ? 22 3 2 3 2 x x y y ? ? ? = ? ? ? ? 22 ( 1) 4 ( 1) 2 ? ? ? ? ? = ? ? ? ? 22 53 ? ? ? = 25 9 ? BC = 34 units................. (2) Distance between the points C(-1,-1) and D(-4,4) is CD = ? ? ? ? 22 4 3 4 3 x x y y ? ? ? = ? ? ? ? 22 ( 4) ( 1) 4 ( 1) ? ? ? ? ? ? = ? ? ? ? 22 35 ?? = 9 25 ? CD = 34 units........................... (3) Distance between the points D (-4,4) and A (1,7) is DA = ? ? ? ? 22 1 4 1 4 x x y y ? ? ? = ? ? ? ? 22 1 ( 4) 7 4 ? ? ? ? = ? ? ? ? 22 53 ? = 25 9 ? DA = 34 units.......................(4) Distance between the points A(1,7) and C(-1, -1) is AC = ? ? ? ? 22 3 1 3 1 x x y y ? ? ? = ? ? ? ? 22 ( 1) 1 ( 1) 7 ? ? ? ? ?Read More

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