Question

The circumcentre of a right triangle always lies:

(Consider taking an example)

• A. inside the triangle
• B. outside the triangle
• C. on its hypotenuse
• D. None of the above

Answer 1

The correct option is C on its hypotenuse

The point of intersection of all the three perpendicular bisectors of a triangle is called the circumcentre of the triangle.

$\underset{–}{\text{Given:}}$ A right triangle.


Let, $AB=p,BC=b$ and $AC=h.$
$DF$ is a perpendicular bisector to $AB.$
$\therefore AD=\frac{p}{2}$

Similarly, $BE=\frac{b}{2}$

As, $\angle B$ is ${90}^o,DF=BE=\frac{b}{2}$

Using Pythagorean theorem, for $△ADF,$

$A{D}^2+D{F}^2=A{F}^2$

$\Rightarrow (\frac{p}{2}{)}^2+(\frac{b}{2}{)}^2=A{F}^2$

$\Rightarrow \frac{p^2}{4}+\frac{b^2}{4}=A{F}^2$

$\Rightarrow \frac{p^2+b^2}{4}=A{F}^2$
$\Rightarrow \frac{h^2}{4}=A{F}^2$
$\Rightarrow (\frac{h}{2}{)}^2=A{F}^2$
$\Rightarrow \frac{h}{2}=AF$

Similarly, considering triangle $△FEC,$ we get $CF=\frac{h}{2}$
$\therefore F$ is the midpoint of $AC.$
Hence, the perpendicular bisetor of $AC$ will pass through $F.$

Also, $F$ is the point of intersection of $DF$ and $FE.$

Hence, it can be proved that the circumcentre of a right triangle always lies on its hypotenuse.