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CBSE Class 10 Answered


The mid points D, E, F of the sides AB, BC, CA of a triangle ABC are (1, 2), (3, -1) and (5, 0). Find the coordinates of the vertices A, B, C of the triangle.

Asked by Topperlearning User | 26 Sep, 2017, 10:33: AM
Expert Answer
begin mathsize 12px style
Given colon space straight D left parenthesis 1 comma space 2 right parenthesis comma space straight E left parenthesis 3 comma space minus 1 right parenthesis space and space straight F left parenthesis 5 comma space 0 right parenthesis
Let space straight A left parenthesis straight x subscript 1 comma space straight y subscript 1 right parenthesis comma space straight B left parenthesis straight x subscript 2 comma space straight y subscript 2 right parenthesis space and space straight C left parenthesis straight x subscript 3 comma space straight y subscript 3 right parenthesis.
straight D comma space straight E space and space straight F space are space the space mid minus point space of space sides space AB comma space BC space and space CA space respectively.
therefore space fraction numerator straight x subscript 1 plus straight x subscript 2 over denominator 2 end fraction equals 1 space and space fraction numerator straight y subscript 1 plus straight y subscript 2 over denominator 2 end fraction equals 2
therefore space straight x subscript 1 plus straight x subscript 2 equals 2 space space space space space space space.... left parenthesis 1 right parenthesis
space space space space space straight y subscript 1 plus straight y subscript 2 equals 4 space space space space space space space space.... left parenthesis 2 right parenthesis
Similarly comma
straight x subscript 2 plus straight x subscript 3 equals 6 space space space space space space space.... left parenthesis 3 right parenthesis
straight y subscript 2 plus straight y subscript 3 equals negative 2 space space space space.... left parenthesis 4 right parenthesis
straight x subscript 1 plus straight x subscript 3 equals 10 space space space space.... left parenthesis 5 right parenthesis
straight y subscript 1 plus straight y subscript 3 equals 0 space space space space.... left parenthesis 6 right parenthesis
Adding space left parenthesis 1 right parenthesis comma space left parenthesis 3 right parenthesis space and space left parenthesis 5 right parenthesis comma
2 left parenthesis straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 right parenthesis equals 18
rightwards double arrow straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 equals 9 space space space space space space.... left parenthesis 7 right parenthesis
Now comma space left parenthesis 7 right parenthesis space minus space left parenthesis 1 right parenthesis space gives space straight x subscript 3 equals 7
left parenthesis 7 right parenthesis space minus space left parenthesis 3 right parenthesis space gives space straight x subscript 1 equals 3
left parenthesis 7 right parenthesis space minus space left parenthesis 5 right parenthesis space gives space straight x subscript 2 equals negative 1
Adding space left parenthesis 2 right parenthesis comma space left parenthesis 4 right parenthesis space and space left parenthesis 6 right parenthesis comma
2 left parenthesis straight y subscript 1 plus straight y subscript 2 plus straight y subscript 3 right parenthesis equals 2
rightwards double arrow straight y subscript 1 plus straight y subscript 2 plus straight y subscript 3 equals 1 space space space space space space.... left parenthesis 8 right parenthesis
Now comma space left parenthesis 8 right parenthesis space minus space left parenthesis 2 right parenthesis space gives space straight y subscript 3 equals negative 3
left parenthesis 8 right parenthesis space minus space left parenthesis 4 right parenthesis space gives space straight y subscript 1 equals 3
left parenthesis 8 right parenthesis space minus space left parenthesis 6 right parenthesis space gives space straight y subscript 2 equals 1
Thus comma space we space have space straight A left parenthesis 3 comma space 3 right parenthesis comma space straight B left parenthesis negative 1 comma space 1 right parenthesis space and space straight C left parenthesis 7 comma space minus 3 right parenthesis. end style
Answered by | 26 Sep, 2017, 12:33: PM
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