The Cartesian plane 15B

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1 Weblink attleship Game e A bishop can move diagonall an number of (unoccupied) squares at a time. omplete the following sentence: With its net move, the bishop at b could capture the (name the chess piece) at (name the position). 7 The game of attleship uses alphanumeric grids. Navigate to the attleship weblink in our eookplus. Pla the game to test our skills with alphanumeric references. 8 7 elesson oordinates and the artesian plane eles-8 REFLETION When we specif an alphanumeric grid reference, does order matter? (That is, does it matter whether the letter is stated first and the number second, or the other wa around?) The artesian plane The artesian plane is named after its inventor, mathematician René Descartes. It is a visual means of describing locations on a plane b using two numbers as coordinates (rather than a letter and a number). The artesian plane is formed b two perpendicular lines. The horizontal line is called the -ais, while the vertical line is referred to as the -ais. The point where the two aes intersect is called the origin. oth aes must be marked (with marks being evenl spaced) and numbered. The distance between each mark is one unit. To locate an point on the artesian plane we use a pair of numbers called artesian coordinates. The numbers are written in a set of brackets and are separated b a comma. The first number in brackets is called the -coordinate of the point; this shows how far across from the origin the point is located. The second number is called the -coordinate; this shows how far up (or down) from the origin the point is. If the artesian coordinates of the point are known, it can be easil located b moving across and up (or down) from the a b c d e f g h origin the specified number of units. For eample, to find the point with coordinates (, ), start from the origin and move units to the right and units up. Hint: To help remember the order in which artesian coordinates are measured, think about using a ladder. Remember we must alwas walk across with our ladder and then climb up it. (, ) hapter oordinates and the artesian plane 7

2 WORKED EXAMPLE Draw a artesian plane with aes etending from to units. Mark the following points with a dot, and label them. a (, ) b (, ) c (, ) d (, ) DRAW First rule up and label the aes. Mark each point. a b c (, ) means starting at the origin, go across units, and then up units. (, ) means go across units and up units. It lies on the -ais. (, ) means go across units and up units. It lies on the -ais. d (, ) means go across units and up unit. Label each point. (, ) (, ) (, ) (, ) WORKED EXAMPLE Find the artesian coordinates for each of the points A,, and D. Point A is units across and unit up Point is unit across and units up. Point is units across and units up. Point D is unit across and units up. WRITE A is at (, ) is at (, ) is at (, ) D is at (, ) D A Etending the aes The artesian aes can etend infinitel in both directions. On the -ais, the values to the left of the origin are negative and decreasing. Likewise, on the -ais the values below the origin are negative and decreasing. The aes divide the artesian plane into four sections called quadrants. The quadrants are numbered in an anti-clockwise direction, starting with the top right corner. If both - and -coordinates of the point are positive, it will be located in the first quadrant; if the -coordinate is negative but the -coordinate is positive, the point will be in the second quadrant. If the point is in the third quadrant, both the - and -coordinates of the point will be negative. Finall, if the point is in the fourth quadrant, its -coordinate is positive, while its -coordinate is negative. 8 Maths Quest 7 for the Australian urriculum nd quadrant st quadrant Origin rd quadrant - - th quadrant - -

3 If the point is located on the -ais, its -coordinate is alwas. Likewise, if the point is on the -ais, its -coordinate is alwas. WORKED EXAMPLE Plot the following points on the artesian plane. A(-, ), (, -), (, -), D(, ), E(-, -) State the location of each point on the plane (that is, the quadrant, or the ais on which it sits). Draw a set of aes, ensuring that the are long enough to fit all the values. Plot the points. The first point is one unit to the left and two units up from the origin. The second point is two units to the right and four units down from the origin (and so on). WRITE A D E Look at the plane and state the location of each point. Remember that the quadrants are numbered in an anticlockwise direction, starting at the top right. If the point is on the ais, specif which ais it is. Point A is in the second quadrant. Point is in the fourth quadrant. Point is on the -ais. Point D is on the -ais. Point E is in the third quadrant. REMEMER. artesian coordinates can be used to locate an point on a plane.. The artesian plane is formed b two perpendicular lines called aes. The horizontal ais is called the -ais and the vertical ais is called the -ais. The aes intersect at the point called the origin.. oth aes must be marked (with marks being evenl spaced) and numbered. The distance between each mark is one unit. The aes can etend infinitel in both directions.. The location of an point on the artesian plane is given b its artesian coordinates. The artesian coordinates are a pair of numbers that are separated b a comma and are shown within brackets. The first number is called the -coordinate of the point; this shows how far across (that is, to the left or to the right) from the origin the point is located. The second number is called the -coordinate; this shows how far up (or down) from the origin nd quadrant st quadrant Origin rd quadrant - - th quadrant - - the point is. For eample, the point (, ) is located units to the right and units up from the origin. hapter oordinates and the artesian plane 9

4 EXERISE INDIVIDUAL PATHWAYS Activit -- Introducing the artesian plane doc-9 Activit -- More of the artesian plane doc- Activit -- D planes doc- The artesian plane FLUENY WE Draw a artesian plane with aes etending from to units. Mark the fol lowing points with a dot, and label them. a (, ) b (, ) c (, ) d (, ) e (, ) f (, ) WE Find the artesian coordinates for each of the points A L L J D G Weblink The coordinate plane I K H F A WE Plot the following points on the artesian plane. A(, ), (, ), (, ), D(, ), E(, ), F(, ), G(, ), H(, ), I(, ), J(, ) State the location of each point on the plane (that is, the quadrant, or the ais it sits on). UNDERSTANDING Each of these sets of artesian aes (ecept one) has something wrong with it. From the list below, match the mistake in each diagram with one of the sentences. A The units are not marked evenl. The -ais is not vertical. The aes are labelled incorrectl. D The units are not marked on the aes. E There is nothing wrong. a b E Maths Quest 7 for the Australian urriculum

5 c d e f Digital docs Spreadsheet Plotting points doc- From the diagram at right, write down the coordinates of points which: a have the same -coordinate b have the same -coordinate. Messages can be sent in code using a grid like the one drawn below, where the letter is represented b the D coordinates (, ). A Use the diagram to decode the answer to the following E riddle. Q Where did the put the man who was run over b a steamroller? A (, )(, )(, )(, )(, )(, )(, )(, )(, ) U V W X Y (, )(, )(, )(, )(, )(, )(, )(, )(, )(, ) P Q R S T (, ) (, )(, )(, )(, )(, )(, )(, )(, )(, ) (, )(, )(, )(, )(, )(, )(, )(, ) K L M N O 7 Rule up a artesian plane with both aes etending from to F G H I J units. Plot the following points and join them in the order A D E given to make a geometric figure. Name each shape. a (, ) (, ) (, ) (, ) b (, ) (8, ) (, 8) (, ) c (, ) (, ) (8, 9) (, 9) (, ) d (, ) (8, ) (, ) (, ) (, ) 8 Here is an eercise which ma require care and concentration. On graph paper or in our eercise book rule up a pair of artesian aes. The -ais must go from to and the -ais from to. Plot the following points and join them in the order given. (, ) (, 7) (9, ) (, ) (, ) (, ) (, ) (8, 9) (, ) (, 8) (, ) (, ) (, ) (, ) (8, ) (, 7) (, 7) (, ) (, 7) (, 8) (, ) N omplete the picture b joining (9, ) (, ) (, ) (9, ). School 9 What is the area of a rectangle formed b connecting the points (, ), (7, ), (7, ) and (, ) on a artesian plane? Home hapter oordinates and the artesian plane

6 Digital docs WorkSHEET. doc- onsider the following set of points: A(, ) (, ) (, 7) D(, ) E(, ) F(, ) G( 8, ) H( 9, ) I(8, 8) J(, ). Which of the following statements is true? a Points A and J are in the first quadrant. b Points and H are in the third quadrant. c Onl point I is in the fourth quadrant. d Onl one point is in the second quadrant. e Point F is at the origin. f Point J is not on the same ais as point E. g Point D is two units to the left of point F. onsider the triangle A at right. a State the coordinates of the vertices of the triangle A. b Find the area of the triangle. A c Reflect the triangle in the -ais. (You need to cop it into our workbook first.) What are the new coordinates of the vertices? d Now reflect the triangle ou have obtained in part c into the -ais, and state the new coordinates of the vertices. REFLETION Wh must the -coordinate alwas be written first and the -coordinate second? Plotting simple linear relationships When a set of points is plotted on the artesian plane, a pattern ma be formed. If a pattern forms a straight line, we call it a linear pattern. The coordinates of the points that form a pattern can be presented as a set, or in a table. If shown in a table (similar to the one shown below), the coordinates of each point should be read in columns ; that is, the top number in each column gives the -coordinate and the bottom number gives the corresponding -coordinate of the point. onsider, for eample, the table of values and the set of points below. oth show the same information. 8 (, 8) (, ) (, ) (, ) The artesian coordinates of the points are ordered pairs. That is, the first number alwas represents the -coordinate, and the second alwas represents the -coordinate of a point. A set of ordered pairs forms a relation between and. If the points form a linear pattern when plotted, we sa that the relation between and is linear. WORKED EXAMPLE Plot the following set of points on the artesian plane, and comment on an pattern formed. (, ) (, ) (, ) (, ) (, 7) WRITE Look at the coordinates of the points in the set: the -values range between and, while the -values range between and 7. Draw a set of aes, ensuring the are long enough to fit all the values. Maths Quest 7 for the Australian urriculum