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Without using the distance formula, show that the points A(4, 5), B(1, 2), C(4, 3) and D(7, 6) are the vertices of a parallelogram.

Straight Line Eq

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Answer

By formula,

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

Without using the distance formula, show that the points A(4, 5), B(1, 2), C(4, 3) and D(7, 6) are the vertices of a parallelogram. Equation of a Line, Concise Mathematics Solutions ICSE Class 10.

⇒Slope of AB=2−51−4=−3−3=1.⇒Slope of CD=6−37−4=33=1.⇒Slope of BC=3−24−1=13.⇒Slope of AD=6−57−4=13.\Rightarrow \text{Slope of AB} = \dfrac{2 - 5}{1 - 4} \\[1em] = \dfrac{-3}{-3} \\[1em] = 1. \\[1em] \Rightarrow \text{Slope of CD} = \dfrac{6 - 3}{7 - 4} \\[1em] = \dfrac{3}{3} \\[1em] = 1. \\[1em] \Rightarrow \text{Slope of BC} = \dfrac{3 - 2}{4 - 1} \\[1em] = \dfrac{1}{3}. \\[1em] \Rightarrow \text{Slope of AD} = \dfrac{6 - 5}{7 - 4} \\[1em] = \dfrac{1}{3}.

From above,

Slope of AB = Slope of CD and Slope of BC = Slope of AD.

∴ AB || CD and BC || AD.

Hence, proved ABCD is a parallelogram.

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