First slide

Cartesian plane

Question

The vertices of a triangle are at O (0, 0), A (a, 0) and B (0, b). The distance between its circumcentre and orthocentre is

Moderate

A

$\sqrt{a^{2} + b^{2}}$
B

$\frac{1}{2} \sqrt{a^{2} + b^{2}}$
C

$\sqrt{\frac{a^{2} + b^{2}}{2}}$
D

$\frac{1}{4} \sqrt{a^{2} + b^{2}}$

Solution

Clearly, $∆ O A B$ is a right triangle right angled at O. So, its orthocentre is at O (0, 0) and circumcentre is the mid- point o its hypotenues i.e. $(\frac{a}{2}, \frac{b}{2})$
$∴$ Required distance $= \sqrt{{(\frac{a}{2} - 0)}^{2} + {(\frac{b}{2} - 0)}^{2}} = \frac{1}{2} \sqrt{a^{2} + b^{2}}$

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