Find x, if \triangle A B C is right-angled at A, where A \equiv \left(\right. 4 , 2 , 3 \left.\right) , B \equiv \left(\right. 3 , 1 , 8 \left.\right) , C \equiv \left(\right. x ,

Engineering

Mathematics

Line of Greatest Slope

Question

Find

x

, if

△ A B C

is right-angled at

A

, where

A \equiv (4, 2, 3), B \equiv (3, 1, 8), C \equiv (x, - 1, 2)

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\begin{array}{l} I n t h e Δ A B C \\ B C^{2} = A C^{2} + A B^{2} \\ {(3 - x)}^{2} + 4 + 36 = 1 + 1 + 25 + {(4 - x)}^{2} + 9 + 1 \\ 9 + 6 x + x^{2} + 40 = 37 + 16 - 8 x + x^{2} \\ 2 x = - 49 + 37 + 16 \\ = - 49 + 53 \\ = 4 \\ x = 2 \end{array}

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