Circumcentre

Q. $A(x_1,y_1),B(x_2,y_2),C(x_3,y_3)$ are the vertices of a triangle. If P (x, y) is the circumcentre of the triangle, then
PA=PB=PC

1. True

2. False

Q. Let $ABC$ be the triangle with $AB=1,AC=3$ and $\angle BAC=\frac{\pi }{2}$. If a circle of radius $r>0$ touches the sides $AB,AC$ and also touches internally the circumcircle of the triangle $ABC,$ then the value of $r$ is

Q. Circumcentre of a triangle, which is formed by the lines y=x, y=2x and y= 3x+4 is

1. (6, 8)
2. (6, -8)
3. (3, 4)
4. (-3, -4)

Q. The mid point of hypotenuse of a right angled triangle is equidistant from all the vertices of the triangle.

1. True
2. False

Q. If $S$ is the circumcentre of $△ABC$, where $A=(1,3),B=(-3,5),C=(5,-1)$, then

1. coordinates of $S$ will be $(-8,-10)$
2. coordinates of $S$ will be $(-4,-5)$
3. radius of circumcircle will be $5\sqrt{10}$ units
4. radius of circumcircle will be $5\sqrt{5}$ units

Q. If $A=(6,0),B=(0,8),$ and the circumcenter of the $△OAB$ where $O$ is the origin, is $(\alpha ,\beta ),$ then the value of $\alpha +\beta$ is

Q. The circumcentre of the triangle formed by $(2,-5),$ $(2,7),$ $(4,7)$ is

1. $(3,1)$
2. $(2,7)$
3. $(3,-1)$
4. $(-3,-1)$

Q. Consider a right angled triangle $ABC$


If the sides of the triangle are $a=6,b=10,c=8$ units, then the distance between incentre and circumcentre will be

1. $\sqrt{5}$
2. $3$
3. $\sqrt{3}$
4. $2$

Q. If $A(x_1,y_1),B{x}_2,y_2),C(x_3,y_3)$ are the vertices of an equilateral triangle and $(x_1-2{)}^2+(y_1-3{)}^2=(x_2-2{)}^2+(y_2-3{)}^2=(x_3-2{)}^2+(y_3-3{)}^2$, then

1. Coordinates of orthocentre is $(0,0)$
2. Coordinates of circumcentre is $(2,3)$
3. value of $(x_1+x_2+x_3)+3(y_1+y_2+y_3)$ is $33$
4. value of $2(x_1+x_2+x_3)-(y_1+y_2+y_3)$ is $3$

Q. If the vertices of a triangle are $P(8,6),Q(8,-2)$ and $R(2,-2)$, then which of the following is/are correct?

1. Coordinates of circumcentre of triangle PQR is $(5,2)$
2. Coordinates of circumcentre of triangle PQR is $(2,5)$
3. Circumradius of triangle PQR is $5$ units
4. Circumradius of triangle PQR is $\sqrt{37}$ units