Distance between Any Two Points on a Plane

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Presentation transcript:

Distance between Any Two Points on a Plane

Well done. Now, try to find the distance between A and B. Do you remember how to calculate the distance between P and Q? x y P( , 5) 1 Q( , 5) B( , 5) 5 A( , 2) 1 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.

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. 2 3 4 1 x y 5 B(5, ) 5 Draw a horizontal line from A and Coordinates of C = ( , ) 5 2 a vertical line from B. AC = (5 – 1) units = 4 units 4 units The two lines intersect at C. C ( , ) 1 5 2 BC = (5 – 2) units = 3 units 3 units A( , 2) By Pythagoras’ theorem, 2 + = BC AC AB AC is a horizontal line. units 3 4 2 + = units 5 = 5 units

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

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

Follow-up question 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 (a) = AB ) 2 5 ( - 2 + ) 1 5 ( - 2 units units 16 9 + = units 25 = units 5 =

Follow-up question 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) = CD 1) 7 ( - 2 + 2)] ( [6 - units 2 units 8 8) ( 2 + - = units 64 + = units) 2 8 (or units 128 =