MathematicsDetermine the vertices of the triangle formed by the lines y=x, 3y=x and x+y=8.

Determine the vertices of the triangle formed by the lines y=x, 3y=x and x+y=8.


  1. A
    (0,0),(0,4),(6,2)
  2. B
    (0,0),(4,4),(0,2)
  3. C
    (0,0),(7,7),(5,2)
  4. D
    (0,0),(4,4),(6,2)  

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    Solution:

    Given equations are y=x, 3y=x and x+y=8.
    We know that the vertices of the triangle will be the point of intersection of given lines taken two at a time.
    So, we need to find the points of intersection of the lines.
    Taking y=x …1,
    3y=x …2
    Let, the point of intersection of above two lines is A.
    Subtract 1 from (2),
    2y=0
    ⇒ y=0
    Put this value of y in equation (1),
    ⇒ x=0
    Hence, the coordinates of A are (0,0).
    Taking y=x ... 2, x+y=8 … (3)
    Let, the point of intersection of above two lines is B.
    Subtract 2 from (3),
    ⇒ x+y-y=8-x
    ⇒ 2x=8
    ⇒ x=4
    Put this value of xin equation (3),
    ⇒ y=4
    Hence, the coordinates of B are (4,4).
    Taking  x+y=8 …3,
    x=3y…1
    Let, the point of intersection of above two lines is C.
    Subtract equation (1) from (3),
    ⇒ x+y-x=8-3y
    ⇒  y=8-3y ⇒ 4y=8
    ⇒ y=2
    Put this value of y in equation (1),
    ⇒ x=3(2)
    ⇒ x=6
    Hence, the coordinates of C are (6,2).
    The coordinates of the triangle are (0,0),(4,4) and (6,2). Hence, the correct option is 4.
     
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