An equilateral triangle has two verteices (3,4) & (-2,3). find the coordinates of the third vertex.

Asked by komal | 16th Jan, 2016, 06:42: PM

Expert Answer:

begin mathsize 14px style L e t space t h e space v e r t i c e s space o f space t h e space e q u i l a t e r a l space t r i a n g l e space b e space A comma space B space a n d space C L e t space A space equals space left parenthesis 3 comma space 4 right parenthesis space space space space space space B space equals space left parenthesis negative 2 comma 3 right parenthesis space a n d space C space equals space left parenthesis x comma space y right parenthesis space space  S i n c e space A B C space i s space a n space e q u i l a t e r a l space t r i a n g l e comma space A C space equals space B C space equals space A B space space  B y space d i s tan c e space f o r m u l a comma A C space equals square root of left parenthesis x minus 3 right parenthesis squared plus left parenthesis y minus 4 right parenthesis squared end root equals space square root of x squared minus 6 x space plus space 9 plus y squared minus 8 y plus 16 end root equals square root of x squared minus 6 x plus y squared minus 8 y plus 25 end root B C space equals space square root of left parenthesis x plus 2 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared end root equals space square root of x squared plus 4 x space plus space 4 plus y squared minus 6 y plus 9 end root equals square root of x squared plus 4 x plus y squared minus 6 y plus 13 end root space A B space equals space square root of left parenthesis negative 2 minus 3 right parenthesis squared plus left parenthesis 3 minus 4 right parenthesis squared end root equals space square root of 26  N o w comma A C space equals space B C space square root of x squared minus 6 x plus y squared minus 8 y plus 25 end root equals square root of x squared plus 4 x plus y squared minus 6 y plus 13 end root space S q u a r i n g space b o t h space s i d e s comma space x squared minus 6 x plus y squared minus 8 y plus 25 space equals space x squared plus 4 x plus y squared minus 6 y plus 13 minus 10 x space plus 12 space equals space 2 y y space equals space 6 minus 5 x  A l s o comma space A C space equals space A B space space square root of x squared minus 6 x plus y squared minus 8 y plus 25 end root space equals square root of 26 S q u a r i n g space b o t h space s i d e s comma x squared minus 6 x plus y squared minus 8 y plus 25 space equals space 26 x squared minus 6 x plus y squared minus 8 y space equals space 1 S u b s t i t u t i n g space y space equals space 6 space minus space 5 x space i n space t h e space a b o v e space e q u a t i o n comma x squared minus 6 x space plus space left parenthesis 6 minus 5 x right parenthesis squared minus 8 left parenthesis 6 minus 5 x right parenthesis space equals space 1 x squared minus 6 x space plus space 36 space minus space 60 x space plus space 25 x squared minus 48 plus 40 x space equals space 1 26 x squared minus 26 x space minus space 13 space equals space 0 2 x squared minus 2 x space minus space 1 space equals space 0 space  S o l v i n g space t h e space a b o v e space q u a d r a t i c space e q u a t i o n comma space w e space g e t comma x space equals space fraction numerator 2 plus-or-minus square root of 12 over denominator 4 end fraction equals fraction numerator 1 plus-or-minus square root of 3 over denominator 2 end fraction T h u s comma space y space equals space 6 space minus space 5 open parentheses fraction numerator 1 plus-or-minus square root of 3 over denominator 2 end fraction close parentheses equals fraction numerator 7 plus-or-minus square root of 3 over denominator 2 end fraction H e n c e comma space c o o r d i n a t e s space o f space t h e space t h i r d space v e r t e x space i s space open parentheses fraction numerator 1 plus square root of 3 over denominator 2 end fraction comma fraction numerator 7 plus square root of 3 over denominator 2 end fraction close parentheses space o r space open parentheses fraction numerator 1 minus square root of 3 over denominator 2 end fraction comma fraction numerator 7 minus square root of 3 over denominator 2 end fraction close parentheses. end style

Answered by Kajal Gupta | 18th Jan, 2016, 01:40: AM

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