Coordinate Geometry

1 CoordinateGeometry8200 Negative gradients: < 0mPositive gradients: > 0mChapter Contents8:01 The distance between two points 8:02 The midpoint of an interval 8:03 The gradient of a line 8:04 Graphing straight lines8:05 The gradient intercept form of a straight line : y = mx + c Investigation: What does y = mx + c tell us?8:06 The equation of a straight line , given point and gradient 8:07 The equation of a straight line , given two points8:08 parallel and perpendicular lines8:09 Graphing inequalities on the number plane Fun Spot: Why did the banana go out with a fig?Mathemathical Terms, Diagnostic Test, Revision Assignment, Working MathematicallyLearning OutcomesStudents will be able to: Find the distance between two points. Find the midpoint of an interval. Find the gradient of an interval. Graph straight lines on the Cartesian plane. Use the gradient-intercept form of a straight line . Find the equation of a straight line given a point and the gradient, or two points on the line .

2 Identify parallel and perpendicular lines. Graph linear inequalities on the Cartesian of InteractionApproaches to Learning (Knowledge Acquisition, Problem Solving, Communication, Logical Thinking, IT Skills, Reflection), Human IngenuityCHAPTER 8 Coordinate GEOMETRY201 The French mathematician Ren Descartes first introduced the number plane. He realised that using two sets of lines to form a square grid allowed the position of a point in the plane to be recorded using a pair of numbers or Geometry is a powerful mathematical technique that allows algebraic methods to be used in the solution of geometrical this chapter, we will look at the basic ideas of: the distance between two points on the number plane the midpoint of an interval gradient (or slope) the relationship between a straight line and its shall then see how these can be used to solve :01 | The Distance Between Two PointsThe number plane is the basis of Coordinate Geometry , an important branch of mathematics.

3 In this chapter, we will look at some of the basic ideas of Coordinate Geometry and how they can be used to solve of the following is the correct statement ofPythagoras theorem for the triangle shown?Aa2 = b2 + c2Bb2 = a2 + c2Cc2 = a2 + b2 For questions 2 to 4, use Pythagoras theorem to find the value of :01abcd cm3 cm4 cmd cm12 cm5 cmd m2 m4 m202 INTERNATIONAL MATHEMATICS 4 Pythagoras theorem can be used to find the distance between two points on the number AB7 Distance AB = .. AB = .. the distance the distance AB = .. 12 3 4 5 6 7 8AB 2 3 1012yAB1 123401 2 3 4 5yx 1 ABAyxB3442yxAB343 1 1 2 3 4101 2 3 4 5yx 1 ABworked examples1 Find the distance between the points (1, 2) and (4, 6).2If A is ( 2, 2) and B is (4, 5) find the length of = a2 + b2c2 = a2 + b2AB2 = AC2 + BC2AB2 = AC2 + BC2= 42 + 32= 32 + 62= 16 + 9= 9 + 36= 25= 45 AB = AB = the length of AB is 5 units. the length of AB is unit. 45 is asurd. Wesimplify surdsif they 12345601 2 3 4 57yx 167A(4, 6)B(1, 2)43C1 12345601 2 3 4 57yx 1 2 3A(4, 5)B( 2, 2)36C254545By drawing a right-angled triangle we can use Pythagoras theorem to find the distance between any two points on the number 8 Coordinate GEOMETRY203 Distance formulaA formula for finding the distance between two points,A(x1, y1) and B(x2, y2), can be found using Pythagoras theorem.

4 We wish to find the length of interval = x2 x1 (since LM = MO LO) (ACML is a rectangle)andRS = y2 y1(since RS = RO SO) (BCSR is a rectangle)NowAB2= AC2 + BC2 (Pythagoras theorem)= (x2 x1)2 + (y2 y1)2 AB = y2 y1x2 x1 RSCLM0xyA(x1, y1)B(x2, y2)y2y1x1x2 OAC = x 2 x 1BC = y 2 y 1x2x1 ()2y2y1 ()2+The distance AB between A(x1, y1) and B(x2, y2) is given by:dx2x1 ()2y2y1 ()2+=worked examples1 Find the distance between the2 Find the distance between thepoints (3, 8) and (5, 4).points ( 2, 0) and (8, 5)Solutions1 Distance = 2 Distance = (x1, y1) = (3, 8) and (x2, y2) = (5, 4)(x1, y1) = ( 2, 0) and (x2, y2) = (8, 5) d = d = = = = = = = Distance 4 47 (using a calculator Distance 11 18 (using a calculatorto answer to 2 decimal places).to answer to 2 decimal places). You should check that the formula will still give the same answer if the coordinates are named in the reverse way. Hence, in example 1, if we call (x1, y1) = (5, 4) and(x2, y2) = (3, 8), we would produce the same ()2y2y1 ()2+x2x1 ()2y2y1 ()2+5 3 ()24 8 ()2+82 ()25 0 ()2+2( )24 ()2+10()25 ()2+416+10025+20125204 INTERNATIONAL MATHEMATICS 4 Use Pythagoras theorem to find the length of each of thefollowing.

5 (Leave your answer as a surd, where necessary.)abcdeFind the lengths BC and AC and use these to find the lengths of AB. (Leave your answers in surd form.)abcdefExercise 8:01 Distance between points1 Use Pythagoras theorem to find the length of the hypotenuse in each of the the distance AB in each of the the length of AB in each of the 4 6 BAyx25 BAyx2424B(5, 5)A(1, 2)yx2 22 3B(3, 2)A( 2, 2)yx 2 Foundation Worksheet 8:01112345678901 2 3 4 5 6 7 886xy10C(7, 1)B(7, 9)A(1, 1)12345678901 2 3 4 5 6 7 843xy10C(3, 3)B(7, 3)A(3, 6)351 1 2 323401 2 3 4 5yx 1B(4, 3)CA(1, 2)521 1 2234501 2 3yx 1 2 3 4B( 3, 3)A(2, 1)C512 1 1 2 3 42301 2 3 4 5yx 1 2 3 4 5 66A( 6, 2)B(6, 3)C21 123401 2 3 4 5yx 1B(5, 4)CA(0, 0)1 123401 2 3 4 5yx 1A(1, 1)B(4, 4)C1 123401yx 1 2 3 4 5 6 7A( 1, 2)B( 7, 4)C1 1 2234501 2 3 4yx 1 2 3B( 3, 2)A(4, 4)Cy1 1 3 223401 2 3 4x 1 2 3B( 2, 3)CA(2, 3) 5 2 3 4 1201 2 3yx 1 2 3 4B( 4, 4)A(2, 1)CCHAPTER 8 Coordinate GEOMETRY205 Use Pythagoras theorem to find the length of interval AB in each of the following.

6 (Leave answers in surd form.)abcdefUse the formula to find the distance between the points:a(4, 2) and (7, 6)b(0, 1) and (8, 7)c( 6, 4) and ( 2, 1)d( 2, 4) and (4, 4)e( 6, 2) and (6, 7)f(4, 9) and ( 1, 3)g(3, 0) and (5, 4)h(8, 2) and (7, 0)i(6, 1) and ( 2, 4)j( 3, 2) and ( 7, 3)k(6, 2) and (1, 1)l(4, 4) and (3, 3)aFind the distance from the point (4, 2) to the of the points ( 1, 2) or (3, 5) is closer to the point (3, 0)?cFind the distance from the point ( 2, 4) to the point (3, 5).dWhich of the points (7, 2) or ( 4, 4) is further from (0, 0)?aThe vertices of a triangle are A(0, 0), B(3, 4) and C( 4, 5).Find the length of each is a parallelogram where A is the point (2, 3), B is (5, 5), C is (4, 3) and D is (1, 1). Show that the opposite sides of theparallelogram are the length of the two diagonals of the parallelogram in part is a quadrilateral, where E is the point (0, 1), F is (3, 2), G is (2, 1) and H is ( 1, 2). Prove that EFGH is a rhombus.

7 (The sides of a rhombus are equal.)e(3, 2) is the centre of a circle. (6, 6) is a point on the circumference. What is the radius of the circle?fProve that the triangle ABC is isosceles if A is ( 2, 1), B is (4, 1) and C is (2, 5).(Isosceles triangles have two sides equal.)gA is the point ( 13, 7) and B is (11, 3). M is halfway between A and B. How far isM from B?31 12345601 2 3 4 5yx 1B(1, 2)A(3, 6)1 1 2234501 2 3 4 5yx 1B(1, 2)A(4, 5)1 1 2234501 2 3 4 5yx 1 2B( 2, 1)A(5, 3)1 1 2234501 2 3 4yx 1 2A( 2, 3)B(3, 1)1 1 2234501 2yx 1 2 3A(1, 3)B( 3, 2)1 1 2 32301 2 3 4 5yx6 7 8A(8, 1)B(2, 1)4dx2x1 ()2y2y1 ()2+=Making a sketchwill MATHEMATICS 48:02 | The Midpoint of an Interval The midpoint of an interval is the halfway M is the midpoint of AB then it will be halfwaybetween A and the the is the average of 4 and 10?4 What is the average of 2 and 4?5 What number is halfwaybetween 4 and 10?6 What number is halfway between 2 and 4?

8 7 What number is9 What number ishalfway betweenhalfway between1 and 5? 1 and 3?810prepquiz8:02410+2---------------24+ 2---------------x1 2 3 4 5 6 7 8 9 10 11 12x 4 3 2 130 1 24 5 6 70123456y 2 101234y15+2------------?=1 3+2-----------------?=If M is the midpointof AB then AM = 2 3 4 5 6 7 8xy9 10B(10, 7)A(4, 3)M(p, q)Formula:px1x2+2----------------=410+2- --------------7=The average of 4and 10 is : 7 is halfway between 4 and :qy1y2+2----------------=37+2----------- -5=The average of 3and 7 is : 5 is halfway between 3 and 8 Coordinate GEOMETRY207 Midpoint formulaUse the graph to find the midpoint of each (p, q)0xyA(x1, y1)B(x2, y2)y2y1x1x2M =Could you pleasesay that in English,Miss?x1 + x2y1 + y2,22 The midpoint, M, of interval AB, whereA is (x1, y1) and B is (x2, y2), is given by:.Mx1x2+2-----------------y1y2+2------ ----------, =worked examples1 Find the midpoint of the interval2 Find the midpoint of interval AB, if A isjoining (2, 6) and (8, 10).

9 The point ( 3, 5) and B is (4, 2).Solutions1 Midpoint = 2 Midpoint = = = = (5, 8)= ( , ) or ( , 1 )x1x2+2----------------y1y2+2----------- -----, x1x2+2----------------y1y2+2------------ ----, 28+2------------610+2---------------, 34+2---------------5 2+2---------------, 12---32---12---12---Exercise 8:02 Midpoint1 Read the midpoint of the intervalAB from the graph2 Find the midpoint of the interval that joins:a(3, 4) to (10, 8)b..3 Find the midpoint of the interval that joins:a( 4, 6) to ( 3, 5)b..Foundation Worksheet 8:0211 123451 2 3yx45(5, 2)(1, 4)1 1 2 321 2 3yx 1 2 3( 3, 1)(1, 3)208 INTERNATIONAL MATHEMATICS 4cdeUse the graph to find the midpointsof the intervals:aABbCDcGHdEFeLMfPQgRShTUiVWFin d the midpoint of each interval AB if:aA is (2, 4), B is (6, 10)bA is (1, 8), B is (5, 6)cA is (4, 1), B is (8, 7)dA is (0, 0), B is ( 4, 2)eA is ( 1, 0), B is (5, 4)fA is ( 2, 6), B is (4, 2)gA is ( 8, 6), B is (0, 10)hA is ( 2, 4), B is ( 4, 6)iA is ( 2, 4), B is ( 6, 7)Find the midpoint of the interval joining:a( 3, 3) and (2, 3)b(8, 1) and (7, 1)c(5, 5) and (5, 5)d(6, 7) and ( 7, 6)e(0, 4) and ( 4, 0)f(6, 6) and (5, 5)g(111, 98) and (63, 42)h(68, 23) and (72, 29)i(400, 52) and (124, 100)a iFind the midpoint of the midpoint of the answers for i and ii the same?

10 IvWhat property of a rectangle does this result demonstrate?1 1 22341 2 3yx 1 2( 2, 4)(2, 2) 1 2 3 4 5 61 2 3yx 1 2(3, 5)( 1, 1)1 1 2231 2 3yx 1 24(4, 2)( 1, 2)6842 2 4 6 8 2 4 62460xyHCDBFELVWRSQPMAUTG234123401 2 3yx45A(1, 3)B(4, 3)D(1, 1)C(4, 1)5 CHAPTER 8 Coordinate GEOMETRY209bIf (4, 6) and (2, 10) are points at opposite ends of a diameterof a circle, what are the coordinates of the centre?c iFind the midpoint of the midpoint of the answers for i and ii the same?ivWhat property of a parallelogram does this result demonstrate?aIf the midpoint of (3, k) and (13, 6) is (8, 3), find the value of midpoint of AB is (7, 3). Find the value of d and e if A is the point (d, 0) andB is ( 1, e).cThe midpoint of AB is ( 6, 2). If A is the point (4, 4), what are the coordinates of B?dA circle with centre (3, 4) has a diameter of AB. If A is the point ( 1, 6) what are the coordinates of B?aIf A is the point (1, 4) and B is the point (15, 10), what are the coordinates of the points C, D and E?