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Mathematics

If A(2, –3), B(–5, 1), C(7, –1) and D(0, k) be four points such that AB is parallel to CD, find the value of k.

Straight Line Eq

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Answer

AB is parallel to line segment CD means they must have the same gradient

By using slope formula,

m = y2−y1x2−x1\dfrac{y2 - y1}{x2 - x1}

Given, points A(2, –3), B(–5, 1)

Substituting values we get,

mAB=1−(−3)−5−2=1+3−7=−47m_{AB} = \dfrac{1 - (-3)}{-5 - 2} = \dfrac{1 + 3}{-7} = -\dfrac{4}{7}

Given, points C(7, –1) and D(0, k)

Substituting values we get,

mCD=k−(−1)0−7=k+1−7m_{CD} = \dfrac{k - (-1)}{0 - 7} = \dfrac{k + 1}{-7}

Equate the Gradients:

⇒−47=k+1−7⇒−47×−7=k+1⇒k+1=4⇒k=4−1⇒k=3.\Rightarrow -\dfrac{4}{7} = \dfrac{k + 1}{-7} \\[1em] \Rightarrow -\dfrac{4}{7} \times -7 = k + 1 \\[1em] \Rightarrow k + 1 = 4 \\[1em] \Rightarrow k = 4 - 1 \\[1em] \Rightarrow k = 3.

Hence, value of k = 3.

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