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  • The vertices of a triangle are <img alt="\mathrm{(0,0),(a, 0) and (0, b).}" src="https://

The vertices of a triangle \mathrm{O A B} are \mathrm{(0,0),(a, 0) and (0, b).} Then, the distance between the orthocentre and circumcentre of the triangle \mathrm{O A B} is
 

Option: 1

\mathrm{a+b}

 


Option: 2

\mathrm{a-b}
 


Option: 3

\mathrm{\frac{1}{2} \sqrt{a^2+b^2}}
 


Option: 4

\mathrm{\frac{1}{2}(a+b)}


Answers (1)

best_answer

For a right-angled triangle \mathrm{A O B}, the orthocentre is \mathrm{O(0,0)} and circumcentre is the middle point of hypotenuse \mathrm{A B.}
\Rightarrow the circumcentre is \mathrm{P\left(\frac{a}{2}, \frac{b}{2}\right)}
Hence, the required distance \mathrm{O P=\sqrt{\frac{a^2}{4}+\frac{b^2}{4}}=\frac{1}{2} \sqrt{a^2+b^2}}

Hence option 3 is correct.
 

Posted by

vinayak

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