Coordinate geometry. Topic 3. Why learn this? What do you know? Learning sequence. number and algebra

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1 Topic 3 Coordinate geometr 3. Overview Wh learn this? What did ou weigh as a bab, and how tall were ou? Did ou grow at a stead (linear) rate, or were there periods in our life when ou grew rapidl? What is the relationship between our height and our weight? We constantl seek to find relationships between variables, and coordinate geometr provides a picture, a visual clue as to what the relationships ma be. What do ou know? Think List what ou know about linear graphs and their equations. Use a thinking tool such as a concept map to show our list. pair Share what ou know with a partner and then with a small group. 3 share As a class, create a large concept map that shows our class s knowledge of linear graphs and their equations. Learning sequence 3. Overview 3. Sketching linear graphs 3.3 Determining linear equations 3. The distance between two points 3. The midpoint of a line segment 3.6 Parallel and perpendicular lines 3.7 Review ONLINE ONLY

2 Watch this video The stor of mathematics: Descartes Searchlight ID: eles-8

3 3. Sketching linear graphs If a series of points (, ) is plotted using the rule = m + c, then the points alwas lie in a straight line whose gradient equals m and whose -intercept equals c. The rule = m + c is called the equation of a straight line written in gradient intercept form. Plotting linear graphs To plot a linear graph, complete a table of values to determine the points. WORKED EXAMPLE TI casio Plot the linear graph defined b the rule = for the -values 3,,,,, and 3. Create a table of values using the given -values. WRITE/DRAW Quadrant Quadrant Quadrant = + Quadrant Find the corresponding -values b substituting each -value into the rule. 3 Plot the points on a Cartesian plane and rule a straight line through them. Since the -values have been specified, the line should onl be drawn between the -values of 3 and (3, ) 3 3 (, ) 3 (, 3) (, ) 6 (, 7) 7 8 (, 9) ( 3, ) 9 = Label the graph. Sketching straight lines A minimum of two points are necessar to plot a straight line. Two methods can be used to plot a straight line: Method : The - and -intercept method. Method : The gradient intercept method. Method : Sketching a straight line using the - and -intercepts As the name implies, this method involves plotting the - and -intercepts, then joining them to sketch the straight line. The line cuts the -ais where = and the -ais where =. 7 Maths Quest + A

4 WORKED EXAMPLE Sketch graphs of the following linear equations b finding the - and -intercepts. a + = 6 b = 3 WRITE/DRAW a Write the equation. a + = 6 Find the -intercept b substituting =. -intercept: when =, + = 6 = 6 = 3 -intercept is (3, ). 3 Find the -intercept b substituting =. -intercept: when =, Plot both points and rule the line. () + = 6 = 6 -intercept is (, 6). + = 6 (, 6) (3, ) Label the graph. b Write the equation. b = 3 Find the -intercept b substituting =. i Add to both sides of the equation. ii Divide both sides of the equation b 3. 3 Find the -intercept. The equation is in the form = m + c, so compare this with our equation to find the -intercept, c. -intercept: when =, 3 = 3 = = -intercept is (, ). c = -intercept is (, ). Plot both points and rule the line. (, ) (, ) = 3 Label the graph. Topic 3 Coordinate geometr 7

5 Method : Sketching a straight line using the gradient intercept method This method is often used if the equation is in the form = m + c, where m represents the gradient (slope) of the straight line, and c represents the -intercept. The steps below outline how to use the gradient intercept method to sketch a linear graph. Step : Plot a point at the -intercept. Step : Write the gradient in the form m = rise. (To write a whole number as a fraction, run place it over a denominator of.) Step 3: Starting from the -intercept, move up the number of units suggested b the rise (move down if the gradient is negative). Step : Move to the right the number of units suggested b the run and plot the second point. Step : Rule a straight line through the two points. WORKED EXAMPLE 3 TI casio Sketch the graph of = 3 using the gradient intercept method. WRITE/DRAW Write the equation of the line. = 3 Identif the value of c (that is, the -intercept) and plot this point. c = 3, so -intercept: (, 3). 3 Write the gradient, m, as a fraction. m = m = rise run, note the rise and run. So, rise = ; run =. Starting from the -intercept at (, 3), move units up and units to the right to find the second point (, ). We have still not found the -intercept (, ) 3 (, 3) Sketching linear graphs of the form = m Graphs given b = m pass through the origin (, ), since c =. A second point ma be determined using the rule = m b substituting a value for to find. WORKED EXAMPLE Sketch the graph of = 3. Write the equation. = 3 Find the - and -intercepts. Note: B recognising the form of this linear equation, = m ou can simpl state that the graph passes through the origin, (, ). WRITE/DRAW -intercept: when =, = 3 = -intercept: (, ) Both the - and -intercepts are at (, ). 76 Maths Quest + A

6 3 Find another point to plot b finding the -value when =. Plot the two points (, ) and (, 3) and rule a straight line through them. When =, = 3 = 3 Another point on the line is (, 3). 3 (, 3) = 3 (, ) Label the graph. Sketching linear graphs of the form = c and = a The line = c is parallel to the -ais, having a gradient of zero and a -intercept of c. The line = a is parallel to the -ais and has an undefined (infinite) gradient. WORKED EXAMPLE Sketch graphs of the following linear equations. a = 3 b = WRITE/DRAW a Write the equation. a = 3 The -intercept is 3. As does not appear in the equation, the line is parallel to the -ais, such that all points on the line have a -coordinate equal to 3. That is, this line is the set of points (, 3) where is an element of the set of real numbers. 3 Sketch a horizontal line through (, 3). -intercept = 3, (, 3) (, 3) = 3 Label the graph. Topic 3 Coordinate geometr 77

7 b Write the equation. b = The -intercept is. As does not appear in the equation, the line is parallel to the -ais, such that all points on the line have an -coordinate equal to. That is, this line is the set of points (, ) where is an element of the set of real numbers. 3 Sketch a vertical line through (, ). -intercept =, (, ) = (, ) Label the graph. Eercise 3. Sketching linear graphs INDIVIDUAL PATHWAYS REFLECTION What tpes of straight lines have an -and -intercept of the same value? Practise Questions:,, 3a h, a e, a d, 6a f, 7a d, 8a d, 9,, consolidate Questions:,, 3f m, a e, a d, 6a f, 7c f, 8a f, 9 master Questions:,, 3h o, d i, c f, 6e i, 7d h, 8c h, 9 3 Individual pathwa interactivit int-7 FLUENCY WE Generate a table of values and then plot the linear graphs defined b the following rules for the given range of -values. Rule -values a = +,, 3,,,, b =,,,, 3, c =. + 6,,,,, d =,,, 3,, e = + 3 3,,,,, f = 7 3,,,,, 78 Maths Quest + A

8 Plot the linear graphs defined b the following rules for the given range of values. Rule a = 3 + b = + 3 c = + 3 values doc-97 doc-98 doc-99 3 WE Sketch graphs of the following linear equations b finding the and intercepts. a 3 = b + 3 = c + 3 = d 3 = e 8 = f + = g + 6 = h + 8 = i + 3 = j + 3 = k 9 + = 36 l 6 = m = n = + o = WE3 Sketch graphs of the following linear equations using the gradient intercept method. a = + b = 3 7 c = + 3 d = e = f = g =.6 +. h = 8 i = 7 WE Sketch the graphs of the following linear equations. a = b = c = 3 d = e = 3 f = 6 WE Sketch the graphs of the following linear equations. a = b = c = d = e = f = g = h = i = 7 Transpose each of the equations to standard form (that is, = m + c). State the and intercept for each. a ( + ) = ( + 3) b ( ) = ( 3) c ( + 3) = 3( + ) d ( ) = ( ) e ( + ) = ( + ) f ( ) = ( + ) g ( + ) = ( ) h 8( ) = ( + 3) i ( +.) = ( 3.) j.( ) = 6.( ) doc- doc- doc- doc-3 UNDERSTANDING 8 Find the and intercepts of the following lines. a = 8 b = c = 9 Eplain wh the gradient of a horizontal line is equal to zero and the gradient of a vertical line is undefined. Topic 3 Coordinate geometr 79

9 REASONING Determine whether 3 = 7 is the equation of a straight line b rearranging 6 into an appropriate form and hence sketch the graph, showing all relevant features. Your friend loves to download music. She earns $ and spends some of it buing music online at $.7 per song. She saves the remainder. Her saving is given b the function f() =.7. a What does represent? b Sketch the function. c How man songs can she bu and still save $? PROBLEM SOLVING A straight line has a general equation defined b = m + c. This line intersects the lines defined b the rules = 7 and = 3. The lines = m + c and = 7 have the same -intercept while = m + c and = 3 have the same -intercept. a On the one set of aes, sketch all three graphs. b Determine the -ais intercept for = m + c. c Determine the gradient for = m + c. d MC The equation of the line defined b = m + c is: A + = 3 B = C = D + = 7 E = 7 3 Water is flowing from a tank at a constant rate. The equation relating the volume of water in the tank, V litres, to the time the water has been flowing from the tank, t minutes, is given b V = 8 t, t. a How much water is in the tank initiall? b Wh is it important that t? c At what rate is the water flowing from the tank? d How long does it take for the tank to empt? e Sketch the graph of V versus t. 3.3 Determining linear equations Finding a linear equation given two points The gradient of a straight line can be calculated from the coordinates of two points (, ) and (, ) that lie on the line. Gradient = m = rise run = The equation of the straight line can then be found in the form = m + c, where c is the intercept. B (, ) Rise = A (, ) Run = 8 Maths Quest + A

10 WORKED EXAMPLE 6 Find the equation of the straight line shown in the graph. 6 3 There are two points given on the straight line: the intercept (3, ) and the intercept (, 6). Find the gradient of the line b appling the formula m = rise run =, where (, ) = (3, ) and (, ) = (, 6). 3 The graph has a intercept of 6, so c = 6. Substitute m =, and c = 6 into = m + c to find the equation. WRITE (3, ), (, 6) m = rise run = = 6 3 = 6 3 = The gradient m =. = m + c = + 6 WORKED EXAMPLE 7 Find the equation of the straight line shown in the graph. (, ) There are two points given on the straight line: the and intercept (, ) and another point (, ). WRITE (, ), (, ) Topic 3 Coordinate geometr 8

11 Find the gradient of the line b appling the formula m = rise run =, where (, ) = (, ) and (, ) = (, ). m = rise run = = = The gradient m =. 3 The intercept is, so c =. Substitute m = and c = into = m + c to determine the equation. = m + c = + = WORKED EXAMPLE 8 TI casio Find the equation of the straight line passing through (, ) and (, ). WRITE Write the general equation of a straight line. = m + c Write the formula for calculating the gradient of a line between two points. 3 Let (, ) and (, ) be the two points (, ) and (, ) respectivel. Substitute the values of the pronumerals into the formula to calculate the gradient. Substitute the value of the gradient into the general rule. Select either of the two points, sa (, ), and substitute its coordinates into = + c. 6 Solve for c; that is, add to both sides of the equation. 7 State the equation b substituting the value of c into = + c. m = m = = 6 3 = = + c Point (, ): = + c = + c = c The equation of the line is = +. Finding the equation of a straight line using the gradient and one point If the gradient of a line is known, onl one point is needed to determine the equation of the line. 8 Maths Quest + A

12 WORKED EXAMPLE 9 Find the equation of the straight line with gradient of and intercept of. Write the known information. The other point is the intercept, which makes the calculation of c straightforward. WRITE Gradient =, -intercept = State the values of m and c. m =, c = 3 Substitute these values into = m + c to find the equation. = m + c = WORKED EXAMPLE TI casio Find the equation of the straight line passing through the point (, ) with a gradient of 3. WRITE Write the known information. Gradient = 3, point (, ). State the values of m, and. m = 3, (, ) = (, ) 3 Substitute the values m = 3, = and = into = m + c and solve to find c. Substitute m = 3 and c = 6 into = m + c to determine the equation. = m + c = 3() + c = + c 6 = c The equation of the line is = 3 6. A simple formula The diagram shows a line of gradient m passing through the point (, ). If (, ) is an other point on the line, then: (, ) m = rise run (, ) m = m( ) = = m( ) The formula = m( ) can be used to write down the equation of a line, given the gradient and the coordinates of one point. Topic 3 Coordinate geometr 83

13 WORKED EXAMPLE Find the equation of the line with a gradient of which passes through the point (3, ). Write the equation in general form, that is in the form a + b + c =. Use the formula = m( ). Write the values of,, and m. WRITE m =, = 3, = = m( ) Substitute for,, and m into the equation. ( ) = ( 3) + = Transpose the equation into the form a + b + c = = + = Eercise 3.3 Determining linear equations INDIVIDUAL PATHWAYS REFLECTION What problems might ou encounter when calculating the equation of a line whose graph is actuall parallel to one of the aes? Practise Questions: a d,, 3,, a d, 7 consolidate Questions: a f,, 3,, c g, 7, 9 Individual pathwa interactivit int-73 master Questions: d h,, 3,, e j, 6 FLUENCY WE6 Determine the equation for each of the straight lines shown. a b c doc-96 doc- d e f g h 7 8 Maths Quest + A

14 WE7 Determine the equation of each of the straight lines shown. a b (, ) 6 (3, 6) 3 c d (, ) ( 8, 6) WE8 Find the equation of the straight line that passes through each pair of points. a (, ) and (3, 6) b (, ) and (3, ) c (, ) and (3, ) d (3, ) and (, ) e (, 6) and (, 6) f ( 3, ) and (, 7) WE9 Find the linear equation given the information in each case below. a Gradient = 3, -intercept = 3 b Gradient = 3, -intercept = c Gradient =, -intercept = d Gradient =, -intercept = e Gradient =, -intercept = f Gradient =., -intercept = g Gradient =, -intercept =. h Gradient = 6, -intercept = 3 i Gradient =., -intercept =. j Gradient = 3., -intercept = 6. WE, For each of the following, find the equation of the straight line with the given gradient and passing through the given point. a Gradient =, point = (, 6) b Gradient =, point = (, 6) c Gradient =, point = (, 7) d Gradient =, point = (8, ) e Gradient = 3, point = (, ) f Gradient = 3, point = (3, 3) g Gradient =, point = (, ) h Gradient =, point = (,.) i Gradient =., point = (6, 6) j Gradient =., point = (, 3) UNDERSTANDING 6 a If t represents the time in hours and C represents cost ($), construct a table of values for 3 hours for the cost of plaing ten pin bowling at the new alle. b Use our table of values to plot a graph of time versus cost. (Hint: Ensure our time ais (horizontal ais) etends to 6 hours and our cost ais (vertical ais) etends to $.) Topic 3 Coordinate geometr 8

15 c i What is the intercept? ii What does the intercept represent in terms of the cost? d Calculate the gradient. e Write a linear equation to describe the relationship between cost and time. f Use our linear equation from part e to calculate the cost of a -hour tournament. g Use our graph to check our answer to part f. REASONING 7 When using the gradient to draw a line, does it matter if ou rise before ou run or run before ou rise? Eplain our answer. 8 a Using the graph below, write a general formula for the gradient m in terms of, and c. b Transpose our formula to make the subject. What do ou notice? (, c) (, ) PROBLEM SOLVING 9 The points A (, ), B (, ) and P (, ) are co-linear. P is a general point that lies anwhere on the line. Show that an equation relating these three points is given b = ( ). B (, ) P (, ) Show that the quadrilateral ABCD is a parallelogram B (3, 6) C (7, 8) A (, ) 3 D (, ) doc-389 A (, ) Maths Quest + A

16 CHALLENGE The distance between two points The distance between two points can be calculated using Pthagoras theorem. Consider two points A (, ) and B (, ) on the Cartesian plane as shown below. A (, ) B (, ) C If point C is placed as shown, ABC is a right angled triangle and AB is the hpotenuse. B Pthagoras theorem: AC = BC = AB = AC + BC = ( ) + ( ) Hence AB = "( ) + ( ) The distance between two points A (, ) and B (, ) is: AB = "( ) + ( ) This distance formula can be used to calculate the distance between an two points on the Cartesian plane. The distance formula has man geometric applications. WORKED EXAMPLE Find the distance between the points A and B in the figure below. Answer correct to two decimal places. A 3 3 B 3 WRITE From the graph, locate points A and B. A( 3, ) and B(3, ) Let A have coordinates (, ). Let (, ) = ( 3, ) Topic 3 Coordinate geometr 87

17 3 Let B have coordinates (, ). Let (, ) = (3, ) Find the length AB b appling the formula for calculating the distance between two points. AB = "( ) + ( ) = "[3 ( 3)] + ( ) = "(6) + (3) =! =! = 3! = 6.7 correct to decimal places Note: If the coordinates were named in the reverse order, the formula would still give the same answer. Check this for ourself using (, ) = (3, ) and (, ) = ( 3, ). WORKED EXAMPLE 3 TI casio Find the distance between the points P (, ) and Q (3, ). WRITE Let P have coordinates (, ). Let (, ) = (, ) Let Q have coordinates (, ). Let (, ) = (3, ) 3 Find the length PQ b appling the formula for the distance between two points. PQ = "( ) + ( ) = "[3 ( )] + ( ) = "() + ( 7) =!6 + 9 =!6 = 8.6 correct to decimal places WORKED EXAMPLE Prove that the points A,, B 3, and C (, 3) are the vertices of an isosceles triangle. Plot the points and draw the triangle. Note: For triangle ABC to be isosceles, two sides must have the same magnitude. WRITE/DRAW A C 3 3 B AC and BC seem to be equal. Find the length AC. A (, ) = (, ) C (, 3) = (, ) AC = "[ ( )] + [ ( 3)] = "() + () =! =! 88 Maths Quest + A

18 3 Find the length BC. B (3, ) = (, ) C (, 3) = (, ) Find the length AB. A, = (, ) B (3, ) = (, ) BC = "[3 ( )] + [ ( 3)] = "() + () =! =! AB = "[3 ()] + [ ()] = "() + ( ) =! + =!8 =! State our proof. Since AC = BC AB, triangle ABC is an isosceles triangle. Eercise 3. The distance between two points INDIVIDUAL PATHWAYS Practise Questions:, a d,, 8, 9 consolidate Questions:, c f,, 7, 9, Individual pathwa interactivit int-7 master Questions:, e h, 3 7, 9 REFLECTION How could ou use the distance formula to show that a series of points la on the circumference of a circle with centre C? FLUENCY WE Find the distance between each pair of points shown at right. 7 G O 6 K B P 3 C E H A N L F I 3 M J D WE3 Find the distance between the following pairs of points. a (, ), (6, 8) b (, ), (, ) c (, 3), ( 7, ) d (, ), (, ) e (, ), (, ) f ( 3, ), (, 3) g (, ), ( 8, ) h (, 7), (, 6) i (a, b), (a, b) j ( a, b), (a, b) UNDERSTANDING 3 MC If the distance between the points (3, b) and (, ) is units, then the value of b is: A 8 B C D E Topic 3 Coordinate geometr 89

19 MC A rhombus has vertices A (, 6), B (6, 6), C (, ) and D (, ). The coordinates of D are: A (, 3) B (, 3) C (, 3) D (3, ) E (3, ) The vertices of a quadrilateral are A (, ), B (, 8), C (, 9) and D (3, ). a Find the lengths of the sides. b Find the lengths of the diagonals. c What tpe of quadrilateral is it? REASONING 6 WE Prove that the points A (, 3), B (, ) and C (, 3) are the vertices of an isosceles triangle. 7 The points P (, ), Q (, ) and R (, 3!3 ) are joined to form a triangle. Prove that triangle PQR is equilateral. 8 Prove that the triangle with vertices D (, 6), E (9, 3) and F, 3) is a right angled triangle. 9 A rectangle has vertices A (, ), B (.6, z), C (7.6, 6.) and D (, ). Find: a the length of CD b the length of AD c the length of the diagonal AC d the value of z. Show that the triangle ABC with coordinates A (a, a), B (m, a) and C ( a, m) is isosceles. PROBLEM SOLVING Triangle ABC is an isosceles triangle where AB = AC, B is the point (, ), C is the point (6, 3) and A is the point (a, 3a). Find the value of the integer constant a. A (a, 3a) C (6, 3) B (, ) ABCD is a parallelogram. a Find the gradients of AB and BC. b Find the coordinates of the point D (, ). c Show that the diagonals AC and BD bisect each other. A (, 6) B (3, 8) C (6, ) doc-6 D (, ) 9 Maths Quest + A

20 3. The midpoint of a line segment Midpoint of a line segment The midpoint of a line segment is the halfwa point. The and coordinates of the midpoint are halfwa between those of the coordinates of the end points. The following diagram shows the line interval AB joining points A (, ) and B (, ). The midpoint of AB is P, so AP = PB. Points C (, ) and D (, ) are added to the diagram and are used to make the two right angled triangles ΔAPC and ΔPBD. The two triangles are congruent: B (, ) P (, ) D A (, ) C AP = PB APC = PBD CAP = DPB So ΔAPC ΔPBD This means that AC = PD; i.e. = (solve for ) i.e. = + (given) (corresponding angles) (corresponding angles) (ASA) = + In other words is simpl the average of and. Similarl, = +. In general, the coordinates of the midpoint of a line segment joining the points (, ) and (, ) can be found b averaging the and coordinates of the end points, respectivel. The coordinates of the midpoint of the line segment joining (, ) M, + ( + ) (, ) (, ) and (, ) are: a +, + b. Topic 3 Coordinate geometr 9

21 WORKED EXAMPLE TI casio Find the coordinates of the midpoint of the line segment joining (, ) and (7, ). WRITE Label the given points (, ) and (, ). Let (, ) = (, ) and (, ) = (7, ) Find the -coordinate of the midpoint. = + = + 7 = = 3 Find the -coordinate of the midpoint. = + = + = 6 = 3 Give the coordinates of the midpoint. The midpoint is Q, 3R. WORKED EXAMPLE 6 The coordinates of the midpoint, M, of the line segment AB are (7, ). If the coordinates of A are (, ), find the coordinates of B. Let the start of the line segment be (, ) and the midpoint be (, ). WRITE/DRAW Let (, ) = (, ) and (, ) = (7, ) The average of the -coordinates is 7. Find the -coordinate of the end point. = + 7 = + = + = 3 3 The average of the coordinates is. Find the -coordinate of the end point. = + = + = + = 8 9 Maths Quest + A

22 Give the coordinates of the end point. The coordinates of the point B are (3, 8). Check that the coordinates are feasible b drawing a diagram. 8 6 B (3, 8) M (7, ) 6 8 A (, ) Eercise 3. The midpoint of a line segment INDIVIDUAL PATHWAYS Practise Questions: a d,, 3a,, 9, consolidate Questions: a d, 6, 9, Individual pathwa interactivit int-7 master Questions: a f, REFLECTION If the midpoint of a line segment is the origin, what are the possible values of the - and -coordinates of the end points? FLUENCY WE Use the formula method to find the coordinates of the midpoint of the line segment joining the following pairs of points. a (, ), (, 8) b (, ), (, ) c (, ), (, ) d (3, ), ( 3, ) e (a, b), (3a, b) f (a + 3b, b), (a b, a b) WE6 The coordinates of the midpoint, M, of the line segment AB are (, 3). If the coordinates of A are (7, ), find the coordinates of B. UNDERSTANDING 3 A square has vertices A (, ), B (, ), C 6, ) and D, ). Find: a the coordinates of the centre b the length of a side c the length of a diagonal. MC The midpoint of the line segment joining the points (, ) and (8, 3) is: A (6, ) B (, ) C (6, ) D (3, ) E (, ) MC If the midpoint of AB is (, ) and the coordinates of B are (3, 8), then A has coordinates: A (, 6.) B (, 3) C (, ) D (, 3) E (7, ) 6 a The vertices of a triangle are A (, ), B, 3) and C (, 3). Find: i the coordinates of P, the midpoint of AC ii the coordinates of Q, the midpoint of AB iii the length of PQ. b Show that BC = PQ. 7 a A quadrilateral has vertices A (6, ), B (, 3), C (, 3) and D (, ). Find: i the midpoint of the diagonal AC ii the midpoint of the diagonal BD. b What can ou infer about the quadrilateral? doc-7 Topic 3 Coordinate geometr 93

23 8 a The points A (, 3.), B (,.) and C ( 6, 6) are the vertices of a triangle. Find: i the midpoint, P, of AB ii the length of PC iii the length of AC iv the length of BC. b Describe the triangle. What does PC represent? REASONING 9 Find the equation of the straight line that passes through the midpoint of A (, ) and B (, 3), and has a gradient of 3. Find the equation of the straight line that passes through the midpoint of A (, 3) and B (3, ), and has a gradient of 3. PROBLEM SOLVING The points A (m, 3m), B (m, m) and C ( 3m, ) are the vertices of a triangle. Show that this is a right-angled triangle. A (m, 3m) C ( 3m, ) B (m, m) doc-38 Write down the coordinates of the midpoint of the line joining the points (3k, k) and (k, 3 k). Show that this point lies on the line with equation + = 9. int Parallel and perpendicular lines Parallel lines Lines that have the same gradient are parallel lines. The three lines on the graph at right all have a gradient of and are parallel to each other. = + = 6 6 = 9 Maths Quest + A

24 WORKED EXAMPLE 7 Show that AB is parallel to CD given that A has coordinates (, ), B has coordinates (, 7), C has coordinates ( 3, ) and D has coordinates (, ). Find the gradient of AB b appling the formula m =. WRITE Let A (, ) = (, ) and B, 7 = (, ) Since m = m AB = 7 ( ) ( ) = 6 = Find the gradient of CD. Let C ( 3, ) = (, ) and D, = (, ) 3 Draw a conclusion. (Note: means is parallel to.) m CD = ( 3) = 7 = Since m AB = m CD =, then AB CD. Collinear points Collinear points are points that all lie on the same straight line. If A, B and C are collinear, then m AB = m BC. B C A WORKED EXAMPLE 8 Show that the points A (, ), B (, ) and C (, ) are collinear. WRITE Find the gradient of AB. Let A (, ) = (, ) and B (, ) = (, ) Since m = m AB = = Topic 3 Coordinate geometr 9

25 Find the gradient of BC. Let B (, ) = (, ) and C (, ) = (, ) m BC = = Show that A, B and C are collinear. Since m AB = m BC = and B is common to both line segments, A, B and C are collinear. = Perpendicular lines There is a special relationship between the gradients of two perpendicular lines. The graph at right shows two perpendicular lines. What do ou notice about their gradients? Consider the diagram shown below, in which the line segment AB is perpendicular to the line segment BC, AC is parallel to the ais, and BD is the perpendicular height of the resulting triangle ABC. In ΔABD, let m AB = m = a b = tan (θ) = B α θ = 6 6 In ΔBCD, let m BC = m a = a c = tan (α) A θ b D c α C In ΔABD, tan(α) = b a So m = b a = m Hence m = m or m m = Hence, if two lines are perpendicular to each other, then the product of their gradients is. Two lines are perpendicular if and onl if: m m = If two lines are perpendicular, then their gradients are a b and b a respectivel. 96 Maths Quest + A

26 WORKED EXAMPLE 9 Show that the lines = + and + = are perpendicular to one another. Find the gradient of the first line. WRITE Hence = + m = Find the gradient of the second line. + = Rewrite in the form = m + c: = = 3 3 Test for perpendicularit. (The two lines are perpendicular if the product of their gradients is.) Hence m = m m = = Hence, the two lines are perpendicular. Determining the equation of a line parallel or perpendicular to another line The gradient properties of parallel and perpendicular lines can be used to solve man problems. WORKED EXAMPLE Find the equation of the line that passes through the point (3, ) and is parallel to the straight line with equation = +. WRITE Write the general equation. = m + c Find the gradient of the given line. The two lines are parallel, so the have the same gradient. 3 Substitute for m in the general equation. Hence so = + has a gradient of m = = + c Substitute the given point to find c. (, ) = (3, ) = (3) + c = 6 + c c = 7 Substitute for c in the general equation. = 7 or 7 = Topic 3 Coordinate geometr 97

27 WORKED EXAMPLE Find the equation of the line that passes through the point (, 3) and is perpendicular to a straight line with a gradient of. For perpendicular lines, m m =. Find the gradient of the perpendicular line. WRITE Given m = m = Use the equation = m ( ) where m = and (, ) = (, 3). Since = m( ) and (, ) = (, 3) then 3 = ( ) 3 = ( 3) = = + = Horizontal and vertical lines Horizontal lines are parallel to the ais, have a gradient of zero, are epressed in the form = c and have no -intercept. Vertical lines are parallel to the ais, have an undefined (infinite) gradient, are epressed in the form = a and have no intercept. = = 6 WORKED EXAMPLE Find the equation of: a the vertical line that passes through the point (, 3) b the horizontal line that passes through the point (, 6). a The equation of a vertical line is = a. The coordinate of the given point is. WRITE a = b The equation of a horizontal line is = c. The coordinate of the given point is 6. b = 6 98 Maths Quest + A

28 WORKED EXAMPLE 3 Find the equation of the perpendicular bisector of the line joining the points (, ) and (6, ). (A bisector is a line that crosses another line at right angles and cuts it into two equal lengths.) Find the gradient of the line joining the given points b appling the formula. m =. WRITE Let (, ) = (, ). Let (6, ) = (, ). m = m = ( ) 6 = 9 6 = 3 Find the gradient of the perpendicular line. m m = m = 3 m = 3 3 Find the midpoint of the line joining the given points. M = a +, + b where (, ) = (, ) and (, ) = (6, ). = + = + 6 = 3 = + = + = Hence 3, are the coordinates of the midpoint. Find the equations of the line with gradient Since = m( ), that passes through 3,. then = 3 ( 3) 3 Simplif b removing the fractions. Multipl both sides b 3. Multipl both sides b. 3 = ( 3) 3 3 = = = Eercise 3.6 Parallel and perpendicular lines INDIVIDUAL PATHWAYS Practise Questions: a d,,, 6a c, 7, 8, 9a c,, 3, 6a b, 8, a,, 3, 6a, 7 consolidate Questions: a d,, 6c d, 7, 8, 9a c,, 3,, 6a b, 7a, 8, a, 3, 6 8, 3, 3 master Questions: c f,, 3,,, 6e f, 7 9, b,,, 3, REFLECTION How could ou use coordinate geometr to design a logo for an organisation? Individual pathwa interactivit int-76 Topic 3 Coordinate geometr 99

29 doc-9 doc- FLUENCY WE7 Find whether AB is parallel to CD given the following sets of points. a A (, 3), B (, 9), C (, ), D (, ) b A,, B 8,, C ( 6, ), D (, 6) c A ( 3, ), B,, C,, D 8, 6 d A (, ), B (, ), C,, D (, ) e A,, B,, C 3,, D 7, 3 f A (, 6), B (, ), C,, D (, ) Which pairs of the following straight lines are parallel? a + + = b = 3 c = 3 d = + 3 e = f 6 = g 3 = + h = 3 WE8 Show that the points A (, ), B, and C (, ) are collinear. Show that the line that passes through the points (, 9) and (, 3) also passes through the point (6, 6). WE9 Show that the lines = 6 3 and = are perpendicular to one another. 6 Determine whether AB is perpendicular to CD, given the following sets of points. a A, 6, B 3, 8, C (, 6), D ( 3, ) b A,, B (, 9), C,, D 7, c A, 3, B, 8, C (, ), D, d A (, ), B,, C,, D (, 8) e A (, 9), B (, 6), C (, 8), D, f A,, B ( 8, ), C ( 6, ), D 3, 7 WE Find the equation of the line that passes through the point (, ) and is parallel to the line with equation =. 8 WE Find the equation of the line that passes through the point (, 7) and is perpendicular to a line with a gradient of 3. 9 Find the equations of the following lines. a Gradient 3 and passing through the point (, ) b Gradient and passing through the point (, ) c Passing through the points (, ) and (, ) d Passing through the points (, 3) and (6, ) e Passing through the point (, ) and parallel to + + = f Passing through the point (, 6) and parallel to 3 = g Passing through the point (, ) and perpendicular to = Find the equation of the line that passes through the point (, ) and is: a parallel to the line with equation 3 = b perpendicular to the line with equation 3 =. Find the equation of the line that contains the point (, ) and is: a parallel to the line with equation 3 = b perpendicular to the line with equation 3 =. Maths Quest + A

30 WE Find the equation of: a the vertical line that passes through the point (, 8) b the horizontal line that passes through the point (, 7). 3 MC a The vertical line passing through the point (3, ) is given b: A = B = 3 C = 3 D = + 3 E = b Which of the following points does the horizontal line given b the equation = pass through? A (, ) B (, ) C (3, ) D (, ) E (, ) c Which of the following statements is true? A Vertical lines have a gradient of zero. B The coordinates of all points on a vertical line are the same. C Horizontal lines have an undefined gradient. D The coordinates of all points on a vertical line are the same. E A horizontal line has the general equation = a. d Which of the following statements is false? A Horizontal lines have a gradient of zero. B The line joining the points (, ) and ( 7, ) is vertical. C Vertical lines have an undefined gradient. D The line joining the points (, ) and ( 7, ) is horizontal. E A horizontal line has the general equation = c. The triangle ABC has vertices A (9, ), B (3, 6), and C (, ). a Find the midpoint, M, of BC. b Find the gradient of BC. c Show that AM is the perpendicular d Describe triangle ABC. bisector of BC. WE3 Find the equation of the perpendicular bisector of the line joining the points (, ) and (, ). 6 Find the equation of the perpendicular bisector of the line joining the points (, 9) and (, ). 7 ABCD is a parallelogram. The coordinates of A, B and C are (, ), (, ) and (, ) respectivel. Find: a the equation of AD b the equation of DC c the coordinates of D. UNDERSTANDING 8 In each of the following, show that ABCD is a parallelogram. a A (, ), B (, 3), C (, ), D (, ) b A (, ), B (, ), C (, 3), D, c A (., 3.), B (, ), C (.,.), D (, ) 9 In each of the following, show that ABCD is a trapezium. a A (, 6), B (, ), C (, ), D (, 9) b A 6, 3, B (8, 6), C (, ), D ( 3, 3) c A (, 7), B (, ), C (.6,.6), D (, 3) Topic 3 Coordinate geometr

31 MC The line that passes through the points (, 6) and (7, 8) also passes through: A (, 3) B (, ) C (, ) D (, 8) E (, ) MC The point (, ) lies on a line parallel to + + =. Another point on the same line as (, ) is: A (, 9) B (, ) C (, ) D (, 3) E (3, ) Find the equation of the straight line given the following conditions. a Passes through the point (, 3) and parallel to = + b Passes through the point (, 3) and parallel to 3 + = 3 3 Determine which pairs of the following lines are perpendicular. a + 3 = b = 7 c = d = + e = 3 + f + 9 = g + = 6 h + = Find the equation of the straight line that cuts the ais at 3 and is perpendicular to the line with equation 3 6 =. Calculate the value of m for which lines with the following pairs of equations are perpendicular to each other. a = 7 and + = m b 6 = 7 and + m = 3 6 MC The gradient of the line perpendicular to the line with equation 3 6 = is: A 3 B 6 C D E 7 MC Triangle ABC has a right angle at B. The vertices are A(, 9), B(, 8) and C(, z). The value of z is: A 8 B C D 7 3 E REASONING 8 The map shows the proposed course for a acht race. Buos have been positioned at A (, ), B (8, 8), C (, 6), and D (, w). a How far is it from the start, O, to buo A? b The race marshall boat, M, is situated halfwa between buos A and C. What are the coordinates of the boat s position? c Stage of the race (from C to D) is perpendicular to stage 3 (from B to C). What is the gradient of CD? d Find the linear equation that describes stage. e Hence determine the eact position of buo D. f An emergenc boat is to be placed at point E, (7, 3). How far is the emergenc boat from the hospital, located at H, km north of the start? 9 Show that the following sets of points form the vertices of a right angled triangle. a A (, ), B (, 3), C (, 7) b A (3, 3), B (, 3), C (, ) c A (, ), B (9, ), C (3, ) 3 Prove that the quadrilateral ABCD is a rectangle when A is (, ), B(6, ), C (3, ) and D (, ) Buo A Scale: unit km M Buo B Buo C E H Buo D O (Start) N Maths Quest + A

32 3 Prove that the quadrilateral ABCD is a rhombus, given A (, 3), B (3, ), C (, 6) and D (, ). Hint: A rhombus is a parallelogram with diagonals that intersect at right angles. 3 a A square has vertices at (, ) and (, ). Where are the other vertices? (There are 3 sets of answers.) b An equilateral triangle has vertices at (, ) and (, ). Where is the other verte? (There are answers.) c A parallelogram has vertices at (, ) and (, ). and (, ). Where is the other verte? (There are 3 sets of answers.) 33 A is the point (, ) and B is the point (, ). a Find the perpendicular bisector of AB. b Show that an point on this line is equidistant from A and B. Questions 3 and 3 relate to the diagram. M is the midpoint of OA. A (, 6) N is the midpoint of AB. 6 P is the midpoint of OB. 3 A simple investigation: a Show that MN is parallel to OB. b Is PN parallel to OA? c Is PM parallel to AB? d Will this be true for an triangle OAB? (Show our proof.) 3 M N 3 A difficult investigation: B a Find the perpendicular bisectors of OA and OB. 3 6 b Find the point W where the two P bisectors intersect. c Show that the perpendicular bisector of AB also passes through W. d Eplain wh W is equidistant from O, A and B. e W is called the circumcentre of triangle OAB. Using W as the centre, draw a circle through O, A and B. PROBLEM SOLVING 36 The lines l and l are at right angles to each other. The line l has the equation p + p + r =. Show that the r distance from M to the origin is given b. "p + p l M l Topic 3 Coordinate geometr 3

33 37 Line A is parallel to the line with equation = 7 and passes through the point (, 3). Line B is perpendicular to the line with equation = and also passes through the point (, 3). Line C intersects with line A where it cuts the -ais and intersects with line B where it cuts the -ais. a Determine the equations for all three lines. Give answers in the form a + b + c =. b Sketch all three lines on the one set of aes. c Determine whether the triangle formed b the three lines is scalene, isosceles or equilateral. Maths Quest + A

34 ONLINE ONLY 3.7 Review The Maths Quest Review is available in a customisable format for students to demonstrate their knowledge of this topic. The Review contains: Fluenc questions allowing students to demonstrate the skills the have developed to efficientl answer questions using the most appropriate methods Problem Solving questions allowing students to demonstrate their abilit to make smart choices, to model and investigate problems, and to communicate solutions effectivel. A summar of the ke points covered and a concept map summar of this topic are available as digital documents. Review questions Download the Review questions document from the links found in our ebookplus. Language int-83 int-833 int-39 aes bisect Cartesian plane collinear coordinates diagonal general form gradient gradient intercept form horizontal linear graph midpoint origin parallel parallelogram perpendicular quadrilateral rhombus rise run segment substitute trapezium vertical vertices intercept intercept Link to assesson for questions to test our readiness FOR learning, our progress AS ou learn and our levels OF achievement. assesson provides sets of questions for ever topic in our course, as well as giving instant feedback and worked solutions to help improve our mathematical skills. The stor of mathematics is an eclusive Jacaranda video series that eplores the histor of mathematics and how it helped shape the world we live in toda. Descartes (eles-8) tells the stor of Rene Descartes, a French philosopher and mathematician who brought the concepts of geometr and algebra together, developing the two-dimensional grid we know as the Cartesian plane. Topic 3 Coordinate geometr

35 number <INVEStigation> investigation and algebra for rich task or <number and algebra> for puzzle rich task What common computer smbol is this? The construction part of this task requires ou to graph nine lines to reveal a common computer smbol. Draw the scale of our graph to accommodate - and -values in the following ranges: 6 and 6. Centre the aes on the grid lines provided. 6 Maths Quest + A

36 Line has the same length as line and is parallel to it. The point (, 3) is the starting point of the line, which decreases in both - and -values from there. Line 6 commences at the same starting point as line, and then runs at right angles to line. It has an -intercept of and is the same length as line. Line 7 commences at the same starting point as both lines and 6. Its equation is = 6 +. The point (, 9) lies at the midpoint. Line 8 has the equation = +. Its midpoint is the point (7, 8) and its etremities are the points where the line meets line 7 and line 9. Line 9 has the equation =. It runs from the intersection of lines and 6 until it meets line 8. What common computer smbol have ou drawn? The top section of our figure is a familiar geometric shape. Use the coordinates on our graph, together with the distance formula to determine the necessar lengths to calculate the area of this figure. 3 Using an smbol of interest to ou, draw our smbol on grid lines and provide instructions for our design. Ensure that our design involves aspects of coordinate geometr that have been used throughout this task. Topic 3 Coordinate geometr 7

37 <INVEStigation> number and algebra for rich task or <number and algebra> for puzzle CODE PUZZLE Who won the inaugural 87 km Sdne to Melbourne marathon in 983? The equations of the straight lines, in the form = m + c, that fit the given information and the letter beside each give the puzzle s answer code. gradient of and -intercept of A gradient of and passes through (, ) C passes through (, ) and (, 3) m = and c = m = and passes through (, 3) D E F gradient of 3 and -intercept of G passes through (, 3) and ( 3, ) H m = and c = 3 I passes through (, 7) and ( 3, ) gradient of and passes through (, ) passes through (, 9) and (, ) L M N m = and c = O gradient of and passes through (, ) P passes through (, ) and (, ) R passes through (3, 8) and has a gradient of passes through (, ) and (3, ) gradient is 3 and passes through (, 7) S T U m = 3 and c = passes through (, 3) with a gradient of passes through (, ) and (, ) V X Y = = 3 = + = = + = = = + = + = + = + = 3 = = + = + 3 = + = = = + = = = + = + = 3 = + 3 = + = 3 = = + = 3 = + = + = + = = 3 + = = 3 + = 3 = = + = 3 = 3 = + = + 3 = + = + = = + = = + 3 = + = 3 = + = = + = + = = 3 = = 3 + = 3 = 8 Maths Quest + A

38 Activities 3. Overview Video The stor of mathematics (eles 8) 3. Sketching linear graphs Interactivit IP interactivit 3. (int-7): Sketching linear graphs Digital docs SkillSHEET (doc 97): Describing the gradient of a line SkillSHEET (doc 98): Plotting a line using a table of values SkillSHEET (doc 99): Stating the intercept from a graph SkillSHEET (doc ): Solving linear equations that arise when finding and intercepts SkillSHEET (doc ): Using Pthagoras theorem SkillSHEET (doc ): Substitution into a linear rule SkillSHEET (doc 3): Transposing linear equations to standard form 3.3 Determining linear equations Interactivit IP interactivit 3.3 (int-73): Determining linear equations Digital docs SkillSHEET (doc 96): Measuring the rise and the run SkillSHEET (doc ): Finding the gradient given two points WorkSHEET 3. (doc 389): Gradient 3. The distance between two points Interactivit IP interactivit 3. (int-7): The distance between two points Digital doc Spreadsheet (doc 6): Distance between two points 3. The midpoint of a line segment Interactivit IP interactivit 3. (int-7): The midpoint of a line segment Digital docs Spreadsheet (doc 7): Midpoint of a segment WorkSHEET 3. (doc 38): Midpoint of a line segment 3.6 Parallel and perpendicular lines Interactivities Parallel and perpendicular lines (int 779) IP interactivit 3.6 (int-76): Parallel and perpendicular lines Digital docs Spreadsheet (doc 9): Perpendicular checker Spreadsheet (doc ): Equation of a straight line 3.7 Review Interactivities Word search (int 83) Crossword (int 833) Sudoku (int 39) Digital docs Topic summar (doc-373) Concept map (doc-37) To access ebookplus activities, log on to Topic 3 Coordinate geometr 9

39 Answers topic 3 Coordinate geometr Eercise 3. Sketching linear graphs a 3 3 b c d = = 3 =. + 6 = 3 e f a b = = = = Maths Quest + A

40 c = + 3 o a = = + (, ) 3 b = (, ) a c e g i k m 3 = + 3 = 8 = h + 6 = + 3 = 9 + = 36 = b d f j l n + 3 = 3 = + = + 8 = = 6 = = c e g i 3 (, ) a = c 3 = + 3 = 3 (, ) = (, 9) (, 3.) = (, 6) = 3 d f h b d 3 3 = (, 9) = (7, ) = (, 8) = 8 3 = Topic 3 Coordinate geometr

41 e 6 a = c = = 3 f 3 e = g i = = b d f = = h = 7 a -intercept:.; -intercept:. b -intercept:.; -intercept:. c -intercept: ; -intercept: d -intercept: 3; -intercept: e -intercept: ; -intercept: f -intercept: ; -intercept:. g -intercept:.7; -intercept:. h -intercept: 7; -intercept: 3. i -intercept: 9.7; -intercept: 3.9 j -intercept: 3.77; -intercept: a (, ), (, 8) b (, ), (, 3) c (, ), (, ) 9 Answers will var. = intercept (,.3) -intercept (3., ) 3 = = a represents the number of songs she bus online. b (, ) 3 =.7 (, 8.7) 3 c a b 7 songs intercept (, 7) = 3 = 7 -intercept (3, ) 3 c 7 3 d B 3 a Initiall there are 8 litres of water. b Time cannot be negative. c litres per minute d minutes e 8 V minutes 6 8 t minutes 6 Eercise 3.3 Determining linear equations a = + b = 3 + c = + d = 8 e = + 3 f = g = 7 h = 3 a = b = 3 c = d = 3 3 a = + 3 b = c = + 7 d = + e = f = 8 a = b = 3 + c = + d = + e = f =. g = +. h = i =. +. j = Maths Quest + A

42 a = 9 b = + 3 c = d = 3 e = 3 3 f = g = + 3 h =. i =. 9 j = a b Cost ($) t 3 C 8 C (6, 38) 36 C = 6t (, ) t 3 6 Time (hours) c i (, ) ii The -intercept represents the initial cost of bowling at the alle, which is the shoe rental. d m = 6 e C = 6t + f $3 g Answers will var. 7 It does not matter if ou rise before ou run or run before ou rise, as long as ou take into account whether the rise or run is negative. 8 a m = c b = m + c 9 Teacher to check m AB = m CD = and m BC = m AD =. As opposite sides have the same gradients, this quadrilateral is a parallelogram. Challenge 3. = 3 + Eercise 3. The distance between two points AB =, CD =! or 6.3, EF = 3! or., GH =! or.7, IJ =, KL =!6 or., MN =! or.66, OP =! or 3.6 a b 3 c d 7.7 e 6.7 f. g 3 h 3 i "a + b j 3"a + b 3 B D a AB =.7, BC =., CD =.7, DA =. b AC =, BD = c Rectangle 6, 7 and 8 Answers will var. 9 a b c 3 d. Answers will var. a = a m AB = and m BC = 7 3 b D (, ) c Teacher to check Eercise 3. The midpoint of a line segment a ( 3, 3 ) b (7, ) c (, ) d (, ) e (a, b) f (a + b, a) ( 3, ) 3 a (3, ) b.7 c 6.3 D C 6 a i (, ) ii (, ) iii 3.9 b BC = 7.8 = PQ 7 a i (,.) ii (,.) b The diagonals bisect each other, so it is a parallelogram. 8 a i (, ) ii 8.9 iii 9. iv 9. b Isosceles. PC is the perpendicular height of the triangle. 9 = = Teacher to check (k, 3. k) Eercise 3.6 Parallel and perpendicular lines a No b Yes c No d No e Yes f No b, f; c, e 3 Answers will var. Answers will var. Answers will var. 6 a Yes b Yes c No d Yes e Yes f No 7 = = 9 a = 3 + b = + 9 c 3 8 = d = e + + = f = g 3 = a + = b + = a 3 + = b = a = b = 7 3 a B b C c D d B a (, ) b c Answers will var. d Isosceles triangle = = 7 a = + b = + 3 c (, ) 8 Answers will var. 9 Answers will var. B E a = + b = 3 a, e; b, f; c, h; d, g = + 3 a m = 8 b m = 8 6 E 7 B 8 a. km b (6.,.) c d = 8 e (, ) f 7.7 km 9, 3, 3 Answers will var. 3 a (, ), (, ) or (, ), (, ) or (, ), (, ) b (,!3 ) or (,!3 ) c (3, ), (, ) or (, ) 33 a = b Answers will var. 3 a Answers will var. b Yes c Yes d Answers will var. Topic 3 Coordinate geometr 3

43 3 a OA: = ; OB: = 3 b (3, 7 ) 3 c, d Answers will var. 36 Teacher to check 37 a Line A: =, Line B: =, Line C: 6 6 = b 3 A (, 3) B (6, ) C (, ) c Scalene Challenge 3. 37, 68,, 3. To find the net number, add the three preceding numbers. Investigation Rich task The smbol is the one used to represent a speaker. The shape is a trapezium. Area = (length line 6 + length line 8) perpendicular distance between these lines. = (! +!) 7! = 6 units 3 Teacher to check Code puzzle Sit one ear old potato farmer Cliff Young in five das and fifteen hours Maths Quest + A

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