Page 1 COORDINATE GEOMETRY 155 7 7.1 Introduction In Class IX, you have studied that to locate the position of a point on a plane, we require a pair of coordinate axes. The distance of a point from the yaxis is called its xcoordinate, or abscissa. The distance of a point from the xaxis is called its ycoordinate, or ordinate. The coordinates of a point on the xaxis are of the form (x, 0), and of a point on the yaxis are of the form (0, y). Here is a play for you. Draw a set of a pair of perpendicular axes on a graph paper. Now plot the following points and join them as directed: Join the point A(4, 8) to B(3, 9) to C(3, 8) to D(1, 6) to E(1, 5) to F(3, 3) to G(6, 3) to H(8, 5) to I(8, 6) to J(6, 8) to K(6, 9) to L(5, 8) to A. Then join the points P(3.5, 7), Q (3, 6) and R(4, 6) to form a triangle. Also join the points X(5.5, 7), Y(5, 6) and Z(6, 6) to form a triangle. Now join S(4, 5), T(4.5, 4) and U(5, 5) to form a triangle. Lastly join S to the points (0, 5) and (0, 6) and join U to the points (9, 5) and (9, 6). What picture have you got? Also, you have seen that a linear equation in two variables of the form ax + by + c = 0, (a, b are not simultaneously zero), when represented graphically, gives a straight line. Further, in Chapter 2, you have seen the graph of y = ax 2 + bx + c (a ? 0), is a parabola. In fact, coordinate geometry has been developed as an algebraic tool for studying geometry of figures. It helps us to study geometry using algebra, and understand algebra with the help of geometry. Because of this, coordinate geometry is widely applied in various fields such as physics, engineering, navigation, seismology and art! In this chapter, you will learn how to find the distance between the two points whose coordinates are given, and to find the area of the triangle formed by three given points. Y ou will also study how to find the coordinates of the point which divides a line segment joining two given points in a given ratio. COORDINATE GEOMETR Y 202021 Page 2 COORDINATE GEOMETRY 155 7 7.1 Introduction In Class IX, you have studied that to locate the position of a point on a plane, we require a pair of coordinate axes. The distance of a point from the yaxis is called its xcoordinate, or abscissa. The distance of a point from the xaxis is called its ycoordinate, or ordinate. The coordinates of a point on the xaxis are of the form (x, 0), and of a point on the yaxis are of the form (0, y). Here is a play for you. Draw a set of a pair of perpendicular axes on a graph paper. Now plot the following points and join them as directed: Join the point A(4, 8) to B(3, 9) to C(3, 8) to D(1, 6) to E(1, 5) to F(3, 3) to G(6, 3) to H(8, 5) to I(8, 6) to J(6, 8) to K(6, 9) to L(5, 8) to A. Then join the points P(3.5, 7), Q (3, 6) and R(4, 6) to form a triangle. Also join the points X(5.5, 7), Y(5, 6) and Z(6, 6) to form a triangle. Now join S(4, 5), T(4.5, 4) and U(5, 5) to form a triangle. Lastly join S to the points (0, 5) and (0, 6) and join U to the points (9, 5) and (9, 6). What picture have you got? Also, you have seen that a linear equation in two variables of the form ax + by + c = 0, (a, b are not simultaneously zero), when represented graphically, gives a straight line. Further, in Chapter 2, you have seen the graph of y = ax 2 + bx + c (a ? 0), is a parabola. In fact, coordinate geometry has been developed as an algebraic tool for studying geometry of figures. It helps us to study geometry using algebra, and understand algebra with the help of geometry. Because of this, coordinate geometry is widely applied in various fields such as physics, engineering, navigation, seismology and art! In this chapter, you will learn how to find the distance between the two points whose coordinates are given, and to find the area of the triangle formed by three given points. Y ou will also study how to find the coordinates of the point which divides a line segment joining two given points in a given ratio. COORDINATE GEOMETR Y 202021 156 MATHEMA TICS 7.2 Distance Formula Let us consider the following situation: A town B is located 36 km east and 15 km north of the town A. How would you find the distance from town A to town B without actually measuring it. Let us see. This situation can be represented graphically as shown in Fig. 7.1. You may use the Pythagoras Theorem to calculate this distance. Now, suppose two points lie on the xaxis. Can we find the distance between them? For instance, consider two points A(4, 0) and B(6, 0) in Fig. 7.2. The points A and B lie on the xaxis. From the figure you can see that OA = 4 units and OB = 6 units. Therefore, the distance of B from A, i.e., AB = OB – OA = 6 – 4 = 2 units. So, if two points lie on the xaxis, we can easily find the distance between them. Now, suppose we take two points lying on the yaxis. Can you find the distance between them. If the points C(0, 3) and D(0, 8) lie on the yaxis, similarly we find that CD = 8 – 3 = 5 units (see Fig. 7.2). Next, can you find the distance of A from C (in Fig. 7.2)? Since OA = 4 units and OC = 3 units, the distance of A from C, i.e., AC = 2 2 3 4 + = 5 units. Similarly, you can find the distance of B from D = BD = 10 units. Now, if we consider two points not lying on coordinate axis, can we find the distance between them? Yes! We shall use Pythagoras theorem to do so. Let us see an example. In Fig. 7.3, the points P(4, 6) and Q(6, 8) lie in the first quadrant. How do we use Pythagoras theorem to find the distance between them? Let us draw PR and QS perpendicular to the xaxis from P and Q respectively. Also, draw a perpendicular from P on QS to meet QS at T. Then the coordinates of R and S are (4, 0) and (6, 0), respectively. So, RS = 2 units. Also, QS = 8 units and TS = PR = 6 units. Fig. 7.1 Fig. 7.2 202021 Page 3 COORDINATE GEOMETRY 155 7 7.1 Introduction In Class IX, you have studied that to locate the position of a point on a plane, we require a pair of coordinate axes. The distance of a point from the yaxis is called its xcoordinate, or abscissa. The distance of a point from the xaxis is called its ycoordinate, or ordinate. The coordinates of a point on the xaxis are of the form (x, 0), and of a point on the yaxis are of the form (0, y). Here is a play for you. Draw a set of a pair of perpendicular axes on a graph paper. Now plot the following points and join them as directed: Join the point A(4, 8) to B(3, 9) to C(3, 8) to D(1, 6) to E(1, 5) to F(3, 3) to G(6, 3) to H(8, 5) to I(8, 6) to J(6, 8) to K(6, 9) to L(5, 8) to A. Then join the points P(3.5, 7), Q (3, 6) and R(4, 6) to form a triangle. Also join the points X(5.5, 7), Y(5, 6) and Z(6, 6) to form a triangle. Now join S(4, 5), T(4.5, 4) and U(5, 5) to form a triangle. Lastly join S to the points (0, 5) and (0, 6) and join U to the points (9, 5) and (9, 6). What picture have you got? Also, you have seen that a linear equation in two variables of the form ax + by + c = 0, (a, b are not simultaneously zero), when represented graphically, gives a straight line. Further, in Chapter 2, you have seen the graph of y = ax 2 + bx + c (a ? 0), is a parabola. In fact, coordinate geometry has been developed as an algebraic tool for studying geometry of figures. It helps us to study geometry using algebra, and understand algebra with the help of geometry. Because of this, coordinate geometry is widely applied in various fields such as physics, engineering, navigation, seismology and art! In this chapter, you will learn how to find the distance between the two points whose coordinates are given, and to find the area of the triangle formed by three given points. Y ou will also study how to find the coordinates of the point which divides a line segment joining two given points in a given ratio. COORDINATE GEOMETR Y 202021 156 MATHEMA TICS 7.2 Distance Formula Let us consider the following situation: A town B is located 36 km east and 15 km north of the town A. How would you find the distance from town A to town B without actually measuring it. Let us see. This situation can be represented graphically as shown in Fig. 7.1. You may use the Pythagoras Theorem to calculate this distance. Now, suppose two points lie on the xaxis. Can we find the distance between them? For instance, consider two points A(4, 0) and B(6, 0) in Fig. 7.2. The points A and B lie on the xaxis. From the figure you can see that OA = 4 units and OB = 6 units. Therefore, the distance of B from A, i.e., AB = OB – OA = 6 – 4 = 2 units. So, if two points lie on the xaxis, we can easily find the distance between them. Now, suppose we take two points lying on the yaxis. Can you find the distance between them. If the points C(0, 3) and D(0, 8) lie on the yaxis, similarly we find that CD = 8 – 3 = 5 units (see Fig. 7.2). Next, can you find the distance of A from C (in Fig. 7.2)? Since OA = 4 units and OC = 3 units, the distance of A from C, i.e., AC = 2 2 3 4 + = 5 units. Similarly, you can find the distance of B from D = BD = 10 units. Now, if we consider two points not lying on coordinate axis, can we find the distance between them? Yes! We shall use Pythagoras theorem to do so. Let us see an example. In Fig. 7.3, the points P(4, 6) and Q(6, 8) lie in the first quadrant. How do we use Pythagoras theorem to find the distance between them? Let us draw PR and QS perpendicular to the xaxis from P and Q respectively. Also, draw a perpendicular from P on QS to meet QS at T. Then the coordinates of R and S are (4, 0) and (6, 0), respectively. So, RS = 2 units. Also, QS = 8 units and TS = PR = 6 units. Fig. 7.1 Fig. 7.2 202021 COORDINATE GEOMETRY 157 Therefore, QT = 2 units and PT = RS = 2 units. Now, using the Pythagoras theorem, we have PQ 2 = PT 2 + QT 2 = 2 2 + 2 2 = 8 So, PQ = 2 2 units How will we find the distance between two points in two different quadrants? Consider the points P(6, 4) and Q(–5, –3) (see Fig. 7.4). Draw QS perpendicular to the xaxis. Also draw a perpendicular PT from the point P on QS (extended) to meet yaxis at the point R. Fig. 7.4 Then PT = 11 units and QT = 7 units. (Why?) Using the Pythagoras Theorem to the right triangle PTQ, we get PQ = 2 2 11 7 + = 170 units. Fig. 7.3 202021 Page 4 COORDINATE GEOMETRY 155 7 7.1 Introduction In Class IX, you have studied that to locate the position of a point on a plane, we require a pair of coordinate axes. The distance of a point from the yaxis is called its xcoordinate, or abscissa. The distance of a point from the xaxis is called its ycoordinate, or ordinate. The coordinates of a point on the xaxis are of the form (x, 0), and of a point on the yaxis are of the form (0, y). Here is a play for you. Draw a set of a pair of perpendicular axes on a graph paper. Now plot the following points and join them as directed: Join the point A(4, 8) to B(3, 9) to C(3, 8) to D(1, 6) to E(1, 5) to F(3, 3) to G(6, 3) to H(8, 5) to I(8, 6) to J(6, 8) to K(6, 9) to L(5, 8) to A. Then join the points P(3.5, 7), Q (3, 6) and R(4, 6) to form a triangle. Also join the points X(5.5, 7), Y(5, 6) and Z(6, 6) to form a triangle. Now join S(4, 5), T(4.5, 4) and U(5, 5) to form a triangle. Lastly join S to the points (0, 5) and (0, 6) and join U to the points (9, 5) and (9, 6). What picture have you got? Also, you have seen that a linear equation in two variables of the form ax + by + c = 0, (a, b are not simultaneously zero), when represented graphically, gives a straight line. Further, in Chapter 2, you have seen the graph of y = ax 2 + bx + c (a ? 0), is a parabola. In fact, coordinate geometry has been developed as an algebraic tool for studying geometry of figures. It helps us to study geometry using algebra, and understand algebra with the help of geometry. Because of this, coordinate geometry is widely applied in various fields such as physics, engineering, navigation, seismology and art! In this chapter, you will learn how to find the distance between the two points whose coordinates are given, and to find the area of the triangle formed by three given points. Y ou will also study how to find the coordinates of the point which divides a line segment joining two given points in a given ratio. COORDINATE GEOMETR Y 202021 156 MATHEMA TICS 7.2 Distance Formula Let us consider the following situation: A town B is located 36 km east and 15 km north of the town A. How would you find the distance from town A to town B without actually measuring it. Let us see. This situation can be represented graphically as shown in Fig. 7.1. You may use the Pythagoras Theorem to calculate this distance. Now, suppose two points lie on the xaxis. Can we find the distance between them? For instance, consider two points A(4, 0) and B(6, 0) in Fig. 7.2. The points A and B lie on the xaxis. From the figure you can see that OA = 4 units and OB = 6 units. Therefore, the distance of B from A, i.e., AB = OB – OA = 6 – 4 = 2 units. So, if two points lie on the xaxis, we can easily find the distance between them. Now, suppose we take two points lying on the yaxis. Can you find the distance between them. If the points C(0, 3) and D(0, 8) lie on the yaxis, similarly we find that CD = 8 – 3 = 5 units (see Fig. 7.2). Next, can you find the distance of A from C (in Fig. 7.2)? Since OA = 4 units and OC = 3 units, the distance of A from C, i.e., AC = 2 2 3 4 + = 5 units. Similarly, you can find the distance of B from D = BD = 10 units. Now, if we consider two points not lying on coordinate axis, can we find the distance between them? Yes! We shall use Pythagoras theorem to do so. Let us see an example. In Fig. 7.3, the points P(4, 6) and Q(6, 8) lie in the first quadrant. How do we use Pythagoras theorem to find the distance between them? Let us draw PR and QS perpendicular to the xaxis from P and Q respectively. Also, draw a perpendicular from P on QS to meet QS at T. Then the coordinates of R and S are (4, 0) and (6, 0), respectively. So, RS = 2 units. Also, QS = 8 units and TS = PR = 6 units. Fig. 7.1 Fig. 7.2 202021 COORDINATE GEOMETRY 157 Therefore, QT = 2 units and PT = RS = 2 units. Now, using the Pythagoras theorem, we have PQ 2 = PT 2 + QT 2 = 2 2 + 2 2 = 8 So, PQ = 2 2 units How will we find the distance between two points in two different quadrants? Consider the points P(6, 4) and Q(–5, –3) (see Fig. 7.4). Draw QS perpendicular to the xaxis. Also draw a perpendicular PT from the point P on QS (extended) to meet yaxis at the point R. Fig. 7.4 Then PT = 11 units and QT = 7 units. (Why?) Using the Pythagoras Theorem to the right triangle PTQ, we get PQ = 2 2 11 7 + = 170 units. Fig. 7.3 202021 158 MATHEMA TICS Let us now find the distance between any two points P(x 1 , y 1 ) and Q(x 2 , y 2 ). Draw PR and QS perpendicular to the xaxis. A perpendicular from the point P on QS is drawn to meet it at the point T (see Fig. 7.5). Then, OR = x 1 , OS = x 2 . So, RS = x 2 – x 1 = PT. Also, SQ = y 2 , ST = PR = y 1 . So, QT = y 2 – y 1 . Now, applying the Pythagoras theorem in ? PTQ, we get PQ 2 = PT 2 + QT 2 = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 Therefore, PQ = ( ) ( ) 2 2 2 1 2 1 x x y y  +  Note that since distance is always nonnegative, we take only the positive square root. So, the distance between the points P(x 1 , y 1 ) and Q(x 2 , y 2 ) is PQ = ( ) ( ) 2 2 2 1 2 1 – + – x x y y , which is called the distance formula. Remarks : 1. In particular, the distance of a point P(x, y) from the origin O(0, 0) is given by OP = 2 2 x y + . 2. We can also write, PQ = ( ) ( ) 2 2 1 2 1 2 x x y y  +  . (Why?) Example 1 : Do the points (3, 2), (–2, –3) and (2, 3) form a triangle? If so, name the type of triangle formed. Solution : Let us apply the distance formula to find the distances PQ, QR and PR, where P(3, 2), Q(–2, –3) and R(2, 3) are the given points. We have PQ = 2 2 2 2 (3 2) (2 3) 5 5 50 + + + = + = = 7.07 (approx.) QR = 2 2 2 2 (–2 – 2) (–3 – 3) (– 4) (– 6) 52 + = + = = 7.21 (approx.) PR = 2 2 2 2 (3 – 2) (2 – 3) 1 ( 1) 2 + = +  = = 1.41 (approx.) Since the sum of any two of these distances is greater than the third distance, therefore, the points P, Q and R form a triangle. Fig. 7.5 202021 Page 5 COORDINATE GEOMETRY 155 7 7.1 Introduction In Class IX, you have studied that to locate the position of a point on a plane, we require a pair of coordinate axes. The distance of a point from the yaxis is called its xcoordinate, or abscissa. The distance of a point from the xaxis is called its ycoordinate, or ordinate. The coordinates of a point on the xaxis are of the form (x, 0), and of a point on the yaxis are of the form (0, y). Here is a play for you. Draw a set of a pair of perpendicular axes on a graph paper. Now plot the following points and join them as directed: Join the point A(4, 8) to B(3, 9) to C(3, 8) to D(1, 6) to E(1, 5) to F(3, 3) to G(6, 3) to H(8, 5) to I(8, 6) to J(6, 8) to K(6, 9) to L(5, 8) to A. Then join the points P(3.5, 7), Q (3, 6) and R(4, 6) to form a triangle. Also join the points X(5.5, 7), Y(5, 6) and Z(6, 6) to form a triangle. Now join S(4, 5), T(4.5, 4) and U(5, 5) to form a triangle. Lastly join S to the points (0, 5) and (0, 6) and join U to the points (9, 5) and (9, 6). What picture have you got? Also, you have seen that a linear equation in two variables of the form ax + by + c = 0, (a, b are not simultaneously zero), when represented graphically, gives a straight line. Further, in Chapter 2, you have seen the graph of y = ax 2 + bx + c (a ? 0), is a parabola. In fact, coordinate geometry has been developed as an algebraic tool for studying geometry of figures. It helps us to study geometry using algebra, and understand algebra with the help of geometry. Because of this, coordinate geometry is widely applied in various fields such as physics, engineering, navigation, seismology and art! In this chapter, you will learn how to find the distance between the two points whose coordinates are given, and to find the area of the triangle formed by three given points. Y ou will also study how to find the coordinates of the point which divides a line segment joining two given points in a given ratio. COORDINATE GEOMETR Y 202021 156 MATHEMA TICS 7.2 Distance Formula Let us consider the following situation: A town B is located 36 km east and 15 km north of the town A. How would you find the distance from town A to town B without actually measuring it. Let us see. This situation can be represented graphically as shown in Fig. 7.1. You may use the Pythagoras Theorem to calculate this distance. Now, suppose two points lie on the xaxis. Can we find the distance between them? For instance, consider two points A(4, 0) and B(6, 0) in Fig. 7.2. The points A and B lie on the xaxis. From the figure you can see that OA = 4 units and OB = 6 units. Therefore, the distance of B from A, i.e., AB = OB – OA = 6 – 4 = 2 units. So, if two points lie on the xaxis, we can easily find the distance between them. Now, suppose we take two points lying on the yaxis. Can you find the distance between them. If the points C(0, 3) and D(0, 8) lie on the yaxis, similarly we find that CD = 8 – 3 = 5 units (see Fig. 7.2). Next, can you find the distance of A from C (in Fig. 7.2)? Since OA = 4 units and OC = 3 units, the distance of A from C, i.e., AC = 2 2 3 4 + = 5 units. Similarly, you can find the distance of B from D = BD = 10 units. Now, if we consider two points not lying on coordinate axis, can we find the distance between them? Yes! We shall use Pythagoras theorem to do so. Let us see an example. In Fig. 7.3, the points P(4, 6) and Q(6, 8) lie in the first quadrant. How do we use Pythagoras theorem to find the distance between them? Let us draw PR and QS perpendicular to the xaxis from P and Q respectively. Also, draw a perpendicular from P on QS to meet QS at T. Then the coordinates of R and S are (4, 0) and (6, 0), respectively. So, RS = 2 units. Also, QS = 8 units and TS = PR = 6 units. Fig. 7.1 Fig. 7.2 202021 COORDINATE GEOMETRY 157 Therefore, QT = 2 units and PT = RS = 2 units. Now, using the Pythagoras theorem, we have PQ 2 = PT 2 + QT 2 = 2 2 + 2 2 = 8 So, PQ = 2 2 units How will we find the distance between two points in two different quadrants? Consider the points P(6, 4) and Q(–5, –3) (see Fig. 7.4). Draw QS perpendicular to the xaxis. Also draw a perpendicular PT from the point P on QS (extended) to meet yaxis at the point R. Fig. 7.4 Then PT = 11 units and QT = 7 units. (Why?) Using the Pythagoras Theorem to the right triangle PTQ, we get PQ = 2 2 11 7 + = 170 units. Fig. 7.3 202021 158 MATHEMA TICS Let us now find the distance between any two points P(x 1 , y 1 ) and Q(x 2 , y 2 ). Draw PR and QS perpendicular to the xaxis. A perpendicular from the point P on QS is drawn to meet it at the point T (see Fig. 7.5). Then, OR = x 1 , OS = x 2 . So, RS = x 2 – x 1 = PT. Also, SQ = y 2 , ST = PR = y 1 . So, QT = y 2 – y 1 . Now, applying the Pythagoras theorem in ? PTQ, we get PQ 2 = PT 2 + QT 2 = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 Therefore, PQ = ( ) ( ) 2 2 2 1 2 1 x x y y  +  Note that since distance is always nonnegative, we take only the positive square root. So, the distance between the points P(x 1 , y 1 ) and Q(x 2 , y 2 ) is PQ = ( ) ( ) 2 2 2 1 2 1 – + – x x y y , which is called the distance formula. Remarks : 1. In particular, the distance of a point P(x, y) from the origin O(0, 0) is given by OP = 2 2 x y + . 2. We can also write, PQ = ( ) ( ) 2 2 1 2 1 2 x x y y  +  . (Why?) Example 1 : Do the points (3, 2), (–2, –3) and (2, 3) form a triangle? If so, name the type of triangle formed. Solution : Let us apply the distance formula to find the distances PQ, QR and PR, where P(3, 2), Q(–2, –3) and R(2, 3) are the given points. We have PQ = 2 2 2 2 (3 2) (2 3) 5 5 50 + + + = + = = 7.07 (approx.) QR = 2 2 2 2 (–2 – 2) (–3 – 3) (– 4) (– 6) 52 + = + = = 7.21 (approx.) PR = 2 2 2 2 (3 – 2) (2 – 3) 1 ( 1) 2 + = +  = = 1.41 (approx.) Since the sum of any two of these distances is greater than the third distance, therefore, the points P, Q and R form a triangle. Fig. 7.5 202021 COORDINATE GEOMETRY 159 Also, PQ 2 + PR 2 = QR 2 , by the converse of Pythagoras theorem, we have ? P = 90°. Therefore, PQR is a right triangle. Example 2 : Show that the points (1, 7), (4, 2), (–1, –1) and (– 4, 4) are the vertices of a square. Solution : Let A(1, 7), B(4, 2), C(–1, –1) and D(– 4, 4) be the given points. One way of showing that ABCD is a square is to use the property that all its sides should be equal and both its digonals should also be equal. Now, AB = 2 2 (1 – 4) (7 2) 9 25 34 +  = + = BC = 2 2 (4 1) (2 1) 25 9 34 + + + = + = CD = 2 2 (–1 4) (–1 – 4) 9 25 34 + + = + = DA = 2 2 (1 4) (7 – 4) 25 9 34 + + = + = AC = 2 2 (1 1) (7 1) 4 64 68 + + + = + = BD = 2 2 (4 4) (2 4) 64 4 68 + +  = + = Since, AB = BC = CD = DA and AC = BD, all the four sides of the quadrilateral ABCD are equal and its diagonals AC and BD are also equal. Thereore, ABCD is a square. Alternative Solution : We find the four sides and one diagonal, say, AC as above. Here AD 2 + DC 2 = 34 + 34 = 68 = AC 2 . Therefore, by the converse of Pythagoras theorem, ? D = 90°. A quadrilateral with all four sides equal and one angle 90° is a square. So, ABCD is a square. Example 3 : Fig. 7.6 shows the arrangement of desks in a classroom. Ashima, Bharti and Camella are seated at A(3, 1), B(6, 4) and C(8, 6) respectively. Do you think they are seated in a line? Give reasons for your answer. Fig. 7.6 202021Read More
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