ABCD is a rectangle, where A,B and C are the points A(3,4) B (1,K) C (4,-3)?

Find the gradient of the line AB giving your answer in terms of K

Determine the two possible values of K

Find the area of the rectangle ABCD for the case in which K is positive

Looking for the method please thank you!!!

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  • 7 years ago
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    (1) By the slope formula, the slope of the line that passes through A and B is just:

    m = Δy/Δx = (k - 4)/(1 - 3) = (k - 4)/(-2) = (4 - k)/2.

    (2) Since ABCD is a rectangle, sides AB and BC must be perpendicular to each other.

    We see that line BC has slope:

    m = Δy/Δx = (-3 - k)/(4 - 1) = (-3 - k)/3.

    We want AB and BC to be perpendicular to each other, so these two slopes (the one we just computed and the one we computed in (1)) must multiply to -1 (since the slope of two perpendicular lines are negative reciprocals).

    So, we require that:

    (-3 - k)/3 * (4 - k)/2 = -1 ==> (-3 - k)(4 - k) = -6, by multiplying both sides by 6

    ==> k^2 - k - 12 = -6, by FOILing the left side

    ==> k^2 - k - 6 = 0, by setting the right side equal to zero

    ==> (k - 3)(k + 2) = 0, by factoring the left side

    ==> k = -2 and k = 3, by the zero-product property.

    (3) I'll leave the details to you, but since we now know the coordinates of B we can compute the area of the rectangle by multiplying the lengths of AB and BC together since AB is either the base of the rectangle or the height (depending on how you look at it---it doesn't matter) and BC is either the height of the base---whichever one isn't AB (in other words, if AB is the base then BC must be the height, and if AB is the height then BC must be the base).

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