MathematicsIf the vertex of a triangle is −8,−17   and the midpoints of the sides through it be 2, 1  and −1, 3  , then find the mid-point of the other two sides.

If the vertex of a triangle is 8,17   and the midpoints of the sides through it be 2, 1  and 1, 3  , then find the mid-point of the other two sides.


  1. A
    x=9&y=21  
  2. B
    x=9&y=12  
  3. C
    x=9&y=21  
  4. D
    x=9&y=21   

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    Solution:

    Given that, the vertex is (-8, -17).
    The midpoints of the sides through it be 2, 1  and 1, 3  .
    Let the coordinate of other two sides be ( x 1 , y 1 ),( x 2 , y 2 )  .
    Let the midpoint of (8,17),( x 1 , y 1 )   be (2, 1) and that of (8,17),( x 2 , y 2 )   be (-1, 3).
    The midpoint between the two points ( x 1 , y 1 ),( x 2 , y 2 )   is given by,
    (x,y)=( x 1 + x 2 2 , y 1 + y 2 2 )  
    Here,
    ( x 1 , y 1 )=(8,17) ( x 2 , y 2 )=( x 1 , y 1 )  
    Since, the midpoint of (8,17),( x 1 , y 1 )   is (2, 1), then,
    2,1 = 8+ x 1 2 , 17+ y 1 2   Equating the x-coordinate,
    8+ x 1 =4 x 1 =12  
    Equating the y-coordinate,
    17+ y 1 =2 y 1 =19   The points are (12, 19).
    Since the midpoint of (8,17),( x 2 , y 2 )   is (-1, 3).
    Here,
    ( x 1 , y 1 )=(8,17) ( x 2 , y 2 )=( x 2 , y 2 )  
    1,3 = 8+ x 2 2 , 17+ y 2 2  
    Equating x-coordinate,
    8+ x 2 =2 x 2 =6  
    Equating y-coordinate,
    17+ y 2 =6 y 2 =23  
    The point is (6, 23).
    Find the midpoint of the vertices (12,19),(6,23)  .
    Substitute the values in the midpoint formula:
    (x,y) 12+6 2 , 19+23 2 = 18 2 , 42 2 = 9,21 x=9&y=21  
    The values are x=9&y=21  .
    Hence, option 3) is correct.
     
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