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Without using the distance formula, show that the points A(4, -2), B(-4, 4) and C(10, 6) are the vertices of a right-angled triangle.

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, -2), B(-4, 4) and C(10, 6) are the vertices of a right-angled triangle. Equation of a Line, Concise Mathematics Solutions ICSE Class 10.

Slope of AB (m1)=4−(−2)−4−4=4+2−8=−68.Slope of AC (m2)=6−(−2)10−4=6+26=86.m1×m2=−68×86=−1.\text{Slope of AB }(m1) = \dfrac{4 - (-2)}{-4 - 4} \\[1em] = \dfrac{4 + 2}{-8} \\[1em] = -\dfrac{6}{8}. \\[1em] \text{Slope of AC }(m2) = \dfrac{6 - (-2)}{10 - 4} \\[1em] = \dfrac{6 + 2}{6} \\[1em] = \dfrac{8}{6}. \\[1em] m1 \times m2 = -\dfrac{6}{8} \times \dfrac{8}{6} = -1.

Since, m1.m2 = -1.

∴ AB ⊥ AC.

Hence, proved that ABC is a right-angled triangle at A.

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