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Standard X

Mathematics

Section Formula

A,B,C,D are f...

Question

$A, B, C, D$ are four points on a plane with position vectors $a, b, c, d,$ respectively, such that $(a - d) . (b - c) = (b - d) . (c - a) = 0.$ For $△ A B C, D$ is the

Incenter

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Orthocenter

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Centroid

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None of these

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Solution

The correct option is A Orthocenter
Since $(a - d) . (b - c) = 0$
$∴ D A . C B = 0$
$∴ A D ⊥ B C .$
Since $(b - d) . (c - a) = 0,$
$∴ D B . A C = 0$
$∴ B D ⊥ C A$
Then, $D$ is the intersection of the altitudes through $A$ and $B$ .
Therefore, $D$ is the orthocenter of the triangle $A B C .$

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A,B,C,D are four points on a plane with position vectors a,b,c,d, respectively, such that (a−d).(b−c)=(b−d).(c−a)=0. For △ABC,D is the

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$A, B, C, D$ are four points on a plane with position vectors $a, b, c, d,$ respectively, such that $(a - d) . (b - c) = (b - d) . (c - a) = 0.$ For $△ A B C, D$ is the