Maths-
General
Easy

Question


In the xy-plane above, if point F (not shown) is placed so that triangle ABC is similar to triangle D F E, which of the following could be the coordinates of the point F ?

  1. (3, -6)
  2. (4, -10)
  3. (6, -7)
  4. (6, -1)

The correct answer is: (4, -10)


    given,ΔABC is similar to  ΔDFE
    Step 1: - find the length respective sides of triangles
    So, fraction numerator A B over denominator D F end fraction equals fraction numerator B C over denominator F E end fraction equals fraction numerator C A over denominator E D end fraction
    AB = square root of left parenthesis 2 minus 1 right parenthesis squared plus left parenthesis 5 minus 1 right parenthesis squared end root equals square root of 1 plus 16 end root equals square root of 17
    BC = square root of left parenthesis 4 minus 2 right parenthesis squared plus left parenthesis 2 minus 5 right parenthesis squared end root equals square root of 4 plus 9 end root equals square root of 13
    CA = square root of left parenthesis 4 minus 1 right parenthesis squared plus left parenthesis 2 minus 1 right parenthesis squared end root equals square root of 9 plus 1 end root equals square root of 10
    DE = square root of left parenthesis 8 minus 2 right parenthesis squared plus left parenthesis negative 4 plus 2 right parenthesis squared end root equals square root of 36 plus 4 end root equals 2 square root of 10
    Step 2:- find the lengths of DF and FE using similar triangles property
    fraction numerator A B over denominator D F end fraction equals fraction numerator B C over denominator F E end fraction equals fraction numerator C A over denominator E D end fraction not stretchy rightwards double arrow fraction numerator square root of 17 over denominator D F end fraction equals fraction numerator square root of 13 over denominator F E end fraction equals fraction numerator square root of 10 over denominator 2 square root of 10 end fraction not stretchy rightwards double arrow fraction numerator square root of 17 over denominator D F end fraction equals fraction numerator square root of 13 over denominator F E end fraction equals 1 half
    D F equals 2 square root of 17 equals square root of 68 equals 8.25
    F E equals 2 square root of 13 equals square root of 52 equals 7.21
    Step 3:- find FE and DF in term of assumed point (a,b)
    Let the point F be (a,b)
    So FE = square root of left parenthesis 8 minus a right parenthesis squared plus left parenthesis negative 4 minus b right parenthesis squared end root equals square root of 52
    So DF = square root of left parenthesis 2 minus a right parenthesis squared plus left parenthesis negative 2 minus b right parenthesis squared end root equals square root of 68
    Step 4:-OPTION checking
    A. (3,-6) =(a,b) for DF = square root of left parenthesis 2 minus 3 right parenthesis squared plus left parenthesis negative 2 plus 6 right parenthesis squared end root equals square root of 1 plus 16 end root equals square root of 17 left parenthesis text  failed condition  end text right parenthesis
    B. (4,-10) =(a,b) for
    DF = square root of left parenthesis 2 minus 4 right parenthesis squared plus left parenthesis negative 2 plus 10 right parenthesis squared end root equals square root of 4 plus 64 end root equals square root of 68 left parenthesis text  success  end text f u l text  condition  end text right parenthesis
    FE = square root of left parenthesis 8 minus 4 right parenthesis squared plus left parenthesis negative 4 plus 10 right parenthesis squared end root equals square root of 16 plus 36 end root equals square root of 52 left parenthesis text  success ful condition  end text right parenthesis

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