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Standard X
Mathematics
Distance Formula
Using the dis...
Question

Using the distance formula, show that the given points are collinear:

(i) (1, -1), (5, 2) and 9, 5 (ii) (6, 9), (0, 1) and (-6, -7)

(iii) (-1, -1), (2, 3) and (8, 11) (iv) (-2, 5), (0, 1) and (2, 3).

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Solution

T h e space d i s tan c e space f o r m u l a space i s space colon square root of left parenthesis x subscript 2 minus x subscript 1 right parenthesis squared plus left parenthesis y subscript 2 minus y subscript 1 right parenthesis squared end root Three space points space left parenthesis say comma A comma space B comma space and space C right parenthesis space are space collinear comma if space the space distance space A C equals distance space A B plus distance space B C left parenthesis space note space colon space in order space to space form space a space triangle comma space the space sum space of space the space lengths space of space two space sides space must space exceed space the space length space of space t he space third space side. space right parenthesis left parenthesis i right parenthesis T h e space d i s tan c e space A C equals square root of left parenthesis 9 minus 1 right parenthesis squared plus left parenthesis 5 minus left parenthesis negative 1 right parenthesis right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 64 plus 36 end root equals square root of 100 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 10 T h e space d i s tan c e space A B equals square root of left parenthesis 5 minus 1 right parenthesis squared plus left parenthesis 2 minus left parenthesis negative 1 right parenthesis right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 16 plus 9 end root space equals square root of 25 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 5 T h e space d i s tan c e space B C equals square root of left parenthesis 9 minus 5 right parenthesis squared plus left parenthesis 5 minus 2 right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 16 plus 9 end root equals square root of 25 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 5 N o w space l e t s space c h e c k space i f space the space distance space A C equals distance space A B plus distance space B C t h a t space i s. comma space w e space h a v e space 10 equals 5 plus 5 space i s space t r u e. T h e r e f o r e space t h e space p o i n t s space A comma B comma C space a r e space c o l l i n e a r.

left parenthesis i i right parenthesis T h e space d i s tan c e space A C equals square root of left parenthesis left parenthesis negative 6 right parenthesis minus 6 right parenthesis squared plus left parenthesis left parenthesis negative 7 right parenthesis minus 9 right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 144 plus 256 end root equals square root of 400 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 20 T h e space d i s tan c e space A B equals square root of left parenthesis 0 minus 6 right parenthesis squared plus left parenthesis 1 minus 9 right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 36 plus 64 end root space equals square root of 100 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 10 T h e space d i s tan c e space B C equals square root of left parenthesis left parenthesis negative 6 right parenthesis minus 0 right parenthesis squared plus left parenthesis left parenthesis negative 7 right parenthesis minus 1 right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 36 plus 64 end root equals square root of 100 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 10 N o w space l e t s space c h e c k space i f space the space distance space A C equals distance space A B plus distance space B C t h a t space i s. comma space w e space h a v e space 20 equals 10 plus 10 space i s space t r u e. T h e r e f o r e space t h e space p o i n t s space A comma B comma C space a r e space c o l l i n e a r.

left parenthesis i i i right parenthesis T h e space d i s tan c e space A C equals square root of left parenthesis 8 minus left parenthesis negative 1 right parenthesis right parenthesis squared plus left parenthesis 11 minus left parenthesis negative 1 right parenthesis right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 81 plus 144 end root equals square root of 225 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 15 T h e space d i s tan c e space A B equals square root of left parenthesis 2 minus left parenthesis negative 1 right parenthesis right parenthesis squared plus left parenthesis 3 minus left parenthesis negative 1 right parenthesis right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 9 plus 16 end root space equals square root of 25 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 5 T h e space d i s tan c e space B C equals square root of left parenthesis 8 minus 2 right parenthesis squared plus left parenthesis 11 minus 3 right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 36 plus 64 end root equals square root of 100 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 10 N o w space l e t s space c h e c k space i f space the space distance space A C equals distance space A B plus distance space B C t h a t space i s. comma space w e space h a v e space 15 equals 5 plus 10 space i s space t r u e. T h e r e f o r e space t h e space p o i n t s space A comma B comma C space a r e space c o l l i n e a r.

left parenthesis i v right parenthesis T h e space d i s tan c e space A C equals square root of left parenthesis 2 minus left parenthesis negative 2 right parenthesis right parenthesis squared plus left parenthesis 3 minus 5 right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 16 plus 4 end root equals square root of 20 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 square root of 5 T h e space d i s tan c e space A B equals square root of left parenthesis 0 minus left parenthesis negative 2 right parenthesis right parenthesis squared plus left parenthesis 1 minus 5 right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 4 plus 16 end root space equals square root of 20 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 square root of 5 T h e space d i s tan c e space B C equals square root of left parenthesis 2 minus 0 right parenthesis squared plus left parenthesis 3 minus 1 right parenthesis squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 4 plus 4 end root equals square root of 8 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 square root of 2 N o w space l e t s space c h e c k space i f space the space distance space A C equals distance space A B plus distance space B C t h a t space i s. comma space w e space h a v e space 2 square root of 5 not equal to 2 square root of 5 plus 2 square root of 2 space. T h e r e f o r e space t h e space p o i n t s space A comma B comma C space a r e space n o t space c o l l i n e a r.

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