BEING TIMED Find the orthocenter of the triangle with the given vertices… K(2, -2), L(4,6), M(8,-2).

Question

BEING TIMED
Find the orthocenter of the triangle with the given vertices…
K(2, -2), L(4,6), M(8,-2).

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Allison 2 months 2021-10-05T11:49:25+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-10-05T11:50:35+00:00

    Answer:

    (5,2)

    Step-by-step explanation:

    you need to go two half way from the farthest point from the left to the right so that would be from 2 to 8 which is a distance of 6 so you go halfway over and that puts you at 5 then you find the distance from the lowest point to the highest point which goes from -2 to 6 with a distance of 8 so that means you go up for which puts you at 2 which means the center of the triangle is at the point (5, 2)

    0
    2021-10-05T11:50:53+00:00

    Answer:

    Therefore orthocentre is ( 5 , 1.5 )

    Step-by-step explanation:

    From the general equation of a circle,

     {x}^{2}  +  {y}^{2}  + 2gx + 2fy + c = 0

    Substitute K(2, -2), L(4,6), M(8,-2) in (x,y)

    K(2, -2): 2^2 +(-2)^2 + 2g(2) + 2f(-2) + c = 0

    4g – 4f + c = -8

    Divide by 4: g f + c = 2 ———–(1)

    L(4,6): 4^2 + 6^2 + 2g(4) + 2f(6) + c = 0

    8g + 12f + c = -52

    Divide by 4: 2g + 3f + c = 13 ———-(2)

    M(8,-2): 8^2 +(-2)^2 + 2g(8) + 2f(-2) + c = 0

    16g – 4f + c = -68

    Divide by 4: 4g f + c = 17 ———-(3)

    Equation (2) (1)

    g + 4f = -11

    g = -11 – 4f ———–(a)

    Equation (3) (2)

    2g – 4f = -4

    2g = -4 + 4f ———-(b)

    Substitute for g in eqn (b)

    2(-11 – 4f) = -4 + 4f

    -12f = 18

    f = 1.5

    From eqn (a)

    g = -11 – 4(-1.5)

    g = -5

    Since the orthocentre is given by (g,f)

    Therefore orthocentre is (5,1.5)

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