The points A(1 , 2), B(3 , 8) and C(x , 20) are collinear if x = *

Question

The points A(1 , 2), B(3 , 8) and C(x , 20) are collinear if x = *

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Everleigh 4 weeks 2021-08-17T18:23:34+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-08-17T18:24:38+00:00

    Given,

    The coordinates of the given points are, A(1 , 2), B(3 , 8) and C(x , 20)

    To find,

    The value of x is the three points are collinear.

    Solution,

    The area of a straight line formed by three points, is always zero. So, we have to apply the area of triangle formula by the given three points,and if for making them collinear we have to assume that the final answer will be zero.

    A = (1,2) = X1,Y1

    B = (3,8) = X2,Y2

    C = (x,20) = X3,Y3

    Area of the triangle :

    ½ × [X1 (Y2-Y3) + X2 (Y3-Y1) + X3 (Y1-Y2)]

    = ½ × [1×(8-20) + 3×(20-2) + x (2-8)]

    = ½ × (-12+54-6x)

    = ½ × (42-6x)

    = (21-3x) square unit

    Now if the three points are collinear, then

    21-3x = 0

    -3x = -21

    3x = 21

    x = 7

    Hence, the value of x will be 7

    0
    2021-08-17T18:24:46+00:00

    SOLUTION

    GIVEN

    The points A(1 , 2), B(3 , 8) and C(x , 20) are collinear

    TO DETERMINE

    The value of x

    EVALUATION

    Here the equation of the line passing through the points A(1 , 2), B(3 , 8) is

     \displaystyle \sf{ \frac{y - 8}{x - 3} =  \frac{8 - 2}{3 - 1}  }

     \implies \displaystyle \sf{ \frac{y - 8}{x - 3} =  \frac{6}{2}  }

     \implies \displaystyle \sf{ \frac{y - 8}{x - 3} =  3}

     \implies \displaystyle \sf{3x - 9 = y - 8}

     \implies \displaystyle \sf{3x - y  =  1} \:  \:  \:  \: .......(1)

    Since the points A(1 , 2), B(3 , 8) and C(x , 20) are collinear

    ∴ C(x , 20) is a point on the line joining the points A(1 , 2), B(3 , 8)

    ∴ C(x , 20) is a point on the line 3x – y = 1

     \therefore \: \sf{3x - 20 = 1}

     \implies \: \sf{3x  = 21}

     \implies \: \sf{x  = 7}

    Hence the required value of x = 7

    FINAL ANSWER

    If the points A(1 , 2), B(3 , 8) and C(x , 20) are collinear then x = 7

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