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Condition of Concurrency of 3 Straight Lines

In Δ ABC, foo...

Question

In $Δ A B C$ , foot of the altitudes from vertices to the opposite sides are (0, 0) (3, 0) and (0, 4) then the orthocenter of $Δ A B C$ is

(0, 0)

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(3, 4)

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(4, 3)

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(1, 1)

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Solution

The correct option is D (1, 1)
The triangle formed by foot of altitudes is called pedal triangle. The in centre of pedal triangle is orthocenter of main triangle

$∴$ Incentre of triangle formed by (0,0) (3,0) (0,4) is ortho centre of $Δ A B C$
$I = (\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c})$
$= (1, 1)$
$= ∴$ orthocenter of $Δ A B C = (1, 1)$

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