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Distance between Any Two Points on a Plane

Distance between Any Two Points on a Plane. y. A ( , 2). 1. x. 0. Well done. Now, try to find the distance between A and B. Do you remember how to calculate the distance between P and Q ?. P ( , 5). 1. B ( , 5). Q ( , 5). 5.

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Distance between Any Two Points on a Plane

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  1. Distance between Any TwoPoints on a Plane

  2. y A( , 2) 1 x 0 Well done. Now, try to find the distance between A and B. Do you remember how to calculate the distance between P and Q? P( , 5) 1 B( , 5) Q( , 5) 5 AB is neither a horizontal line nor a vertical line. I don’t know how to calculate the distance. The distance between P and Q is (5  1) units = 4 units.

  3. y 5 4 3 2 1 x 0 1 2 3 4 5 2 2 = + AB AC BC 2 2 = + 4 3 units = 5 units Distance between Any Two Points on a Plane BC is a vertical line. Consider two points A(1, 2) and B(5, 5) on a rectangular coordinate plane. B(5, ) 5 ( , ) Coordinates of C = 2 5 Draw a horizontal line from A and a vertical line from B. 4 units AC = (5 – 1) units = 4 units The two lines intersect at C. ( , ) C 5 2 1 A( , 2) BC = (5 – 2) units = 3 units 3 units By Pythagoras’ theorem, AC is a horizontal line. 5 units

  4. y B(x2, y2) y2 – y1 x 0 A(x1, y1) x2 – x1 ( ) ( ) - 2 2 = - y y + AB x x 2 1 2 1 In general, for any two points A(x1, y1) and B(x2, y2) on a rectangular coordinate plane, x2 y1 C(, ) It is known as the distance formulabetween two points.

  5. = PQ - ( 3)] = 2 12 = = 169 units = 13 units y Find the length of PQ in the figure. Q( , ) 6 9 3 1 P( , ) x 0 2 2 + - - [9 (6 1) units Remember to write the ‘units’. 2 + 5 units + 144 25 units

  6. = - - AB ( 5 2 ) ( 5 1 ) = + 9 16 units = 25 units = 5 units Follow-up question 1 In each of the following, find the distance between the two given points. (a)A(2, 1) and B(5, 5) (b)C(1, 2) and D(7, 6) (Leave your answers in surd form if necessary.) Solution 2 2 (a) + units

  7. 2 2 = - - + - - CD ( 7 1) [6 ( 2)] units 2 2 = - + ( 8) 8 units = + 64 64 units = 128 units (or 8 2 units) Follow-up question 2 In each of the following, find the distance between the two given points. (a)A(2, 1) and B(5, 5) (b)C(1, 2) and D(7, 6) (Leave your answers in surd form if necessary.) Solution (b)

  8. Example 1

  9. Solution

  10. Example 2

  11. Solution

  12. Example 3

  13. Solution

  14. Slope and Inclination of a Straight Line

  15. y x 0 You are right! In fact, in coordinate geometry, we use or to describe the steepness of a straight line. How about the steepness of these two lines? Let’s consider the two paths below. Which path is steeper? slope inclination Straight line B Straight line A Path B Path A It seems that straight line B is steeper. Of course, path B is steeper.

  16. y B slope of a straight line vertical change vertical change = x 0 A horizontal change Slope of a Straight Line The slope of a straight line is the ratio of the vertical change to the horizontal change between any two points on the straight line, i.e. horizontal change

  17. Slope of a straight line vertical change = horizontal change - y y = 2 1 - x x 2 1 Consider a straight line L passing through A(x1, y1) and B(x2, y2), where x1x2. Coordinates of C = (x2, y1) y L y2 x2 B( , ) vertical change x 0 y1 x1 A(, ) y1 x2 C (, ) horizontal change

  18. y L B(x2, y2) x 0 A(x1, y1) - + y y - (y y ) 1 2 1 2 m = - + - (x x x x ) 1 2 1 2 - - - y y y y y y 1 2 2 1 1 2 - m m m   = y y 1 2 - - - x x x x x x 1 2 2 1 2 1 - x x 1 2 If we use the letter m to represent the slope of the straight line L, then or and Note:

  19. B(4, 3) y - y = 2 1 Slope of AB x 0 x - x A(–1, –1) 2 1 - - 3 ( 1) = (x1, y1) = (1, 1) (x2, y2)= (4, 3) - - 4 ( 1) 4 = 5 Let’s find the slope of AB. y

  20. y B(4, 3) x 0 A(–1, –1) = slope of AB y - y 1 2 x - x - - 1 3 1 2 = (x1, y1) = (1, 1) (x2, y2)= (4, 3) - - 1 4 4 = 5 Let’s find the slope of AB. Alternatively,

  21. The slopes of AB and CD are in opposite sign. - 5 1 = Slope of CD - 3 1 - - 4 ( 2 ) = 2 = Slope of AB = - 6 - 2 3 Follow-up question 3 In each of the following, find the slope of the straight line passing through the two given points. (a)A(2, 4) and B(3, –2) (b)C(1, 1) and D(3, 5) Solution (b) (a) What does this mean?

  22. Slope = 2 y y Slope = 1 Slope = 0.5 Slope = –2 Slope = –1 x x 0 0 Slope = –0.5 In fact, for straight lines sloping upwards from left to right, their slopes are positive. for straight lines sloping downwards from left to right, their slopes are negative. Slope Slope < 0 > 0

  23. Slope = 2 y y Slope = 1 for straight lines sloping upwards from left to right, their slopes are positive. Slope = 0.5 Slope = –2 Slope = –1 x x 0 0 for straight lines sloping downwards from left to right, their slopes are negative. Slope = –0.5 In fact, The steepest line The steepest line Slope > 0 Slope < 0 The greater the numericalvalue of the slope, the steeper is the straight line. The greater the value of the slope, the steeperis the straight line. 2 1 0.5 > > 2 1 0.5 > >

  24. What are the slopes of a horizontal line and a vertical line?

  25. The slope of a horizontal line is . y y B(x2, y1) A(x1, y1) - y y 1 1 = Slope of AB - x x 2 1 = 0 x x 0 0 - y y 2 1 = 0 D(x1, y1) - y y 2 1 = Slope of CD - x x 1 1 C(x1, y2) 0 For a line that is parallel to the x-axis, 2. The slope of a vertical line is . undefined For a line that is parallel to the y-axis,  It is meaningless to divide a number by 0.

  26. y x 0 Follow-up question 4 On the rectangular coordinate plane as shown, L1, L2, L3 and L4 are four straight lines. Given that their slopes are 0, 0.5, 1 and 2 (not in the corresponding order), determine the slopes of each line according to their steepness. L1 L2 L3 2 1 0.5 0 L4 L1 and L2 are sloping upwards from left to right and L1 is steeper. L4 is sloping downwards from left to right. L3 is a horizontal line.

  27. Inclination We can also describe the steepness of a straight line by its inclination. y  is the angle that the straight line L makes with the positive x-axis (measured anti-clockwise from the x-axis to L) Straight line L  x positive x-axis 0  is called the inclination of L. Note: For 0 <  < 90, when  increases, the steepness of L also increases.

  28. Is there any relationship between the inclination of a straight line and its slope?

  29. y L B a A  x 0 Consider a straight line L passing through A and B with inclination  . Draw a horizontal line from A and C a vertical line from B. They intersect at C. Let BAC = a.

  30. y B a A  x BC BC 0 Slope Slope of of L = L = AC AC ∴ q q = tan tan tan a BC AC Consider a straight line L passing through A and B with inclination  . L C =a ∵  and aare corresponding angles. Note that ACB = 90. = By the definition of tangent ratio ∴ = tan 

  31. y L 50 0 The relationship between the inclination  and the slope of a straight line L is slope of L = tan  For example: If the inclination of a straight line L is 50, slope of L = tan 50 = 1.19 (cor. to 3 sig. fig.) x

  32. ∵ Slope of L = tan  ∴ = tan  q =  60 Let’s find the inclination  of L. y L slope = 60 x 0

  33. = . 700 (cor. to 3 sig. fig.) 0 Follow-up question 5 (a) Given that the inclination of a straight line L is 35, find the slope of L correct to 3 significant figures. Solution (a) Slope of L = tan 35

  34. q =  63 Follow-up question 5 (b) Given that the slope of a straight line L is 2, find the inclination  of L correct to the nearest degree. Solution (b) Slope of L = tan  2 = tan  ∴ (cor. to the nearest degree)

  35. Example 4 Solution

  36. Example 5 Solution

  37. Example 6 Solution

  38. Example 7 Solution

  39. Example 8 Solution

  40. Example 9 Solution

  41. Example 10 Solution

  42. Parallel and Perpendicular Lines

  43. Yes. If now, we rotate the lines to the same extent, what do you think about their steepness and their slopes? The figure shows two parallel horizontal lines. What are their slopes? Good. Actually, the slopes of parallel lines are always equal. Let me show you the proof. y slope = 0 slope = 0 x 0 By observation, it seems that their steepness are always the same, so they have the same slopes. Both of the slopes are 0.

  44. Parallel Lines y The figure shows two straight lines L1 and L2, whose inclinations are 1 and 2 respectively. L1 L2 1 2 If L1 // L2, then ∴ i.e. slope of L1= slope of L2 x 0 1 = 2 (corr. s, L1 // L2) tan 1 = tan 2

  45. From the above result, we have If L1 // L2, then m1 = m2. The converse of the above result is also true: If m1 = m2, thenL1 // L2.

  46. - 6 3 = Slope of AB - - 1 ( 1 ) 3 3 = = 2 2 - 5 2 B(1, 6) = Slope of CD - 8 6 D(8, 5) A(–1, 3) C(6, 2) Determine whether two lines AB and CD are parallel. y x 0 ∵ Slope of AB = slope of CD ∴ AB // CD

  47. - - 6 5 11 - = = Slope of RS - 6 2 4 - - 8 3 11 - = = Slope of PQ - - - 2 ( 6) 4 Follow-up question 6 The figure shows four points P(6, 3), Q(2, 8), R(2, 5) and S(6, 6). Prove that PQ is parallel to RS. y R(2, 5) P(6, 3) x 0 Solution S(6, 6) Q(2, 8) ∵ Slope of PQ = slope of RS ∴ PQ // RS

  48. Example 11 Solution

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