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Orthocentre

If the orthoc...

Question

If the orthocentre of the triangle formed by the points $A (2, 0),$ $B (2, 3)$ and $C (0, 3)$ is $(α, β),$ then the value of $α + β$ is

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Solution

Given vertices are $A (2, 0), B (2, 3)$ and $C (0, 3)$
Now, the side lengths are
$A B = \sqrt{0^{2} + 3^{2}} = 3$ units
$B C = \sqrt{2^{2} + 0} = 2$ units
$A C = \sqrt{2^{2} + 3^{2}} = \sqrt{13}$ units

By observation, we see that
$A C^{2} = A B^{2} + B C^{2}$
Given triangle is right-angled at vertex $B$
So, the orthocentre is $(α, β) = (2, 3)$
$∴ α + β = 2 + 3 = 5$

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