Distance Between Two Points

Show that $\left(2, 1\right)$ is the centre of the circumcircle of the triangle whose vertices are $\left(-3, -9\right), \left(13, -1\right)$ and $\left(-9, 3\right)$.

The coordinates of $A$ are $\left(-4, 8\right)$ and those of $B$ are $\left(x, 3\right)$. Find $x$ if $AB=13$ unit.

Show that the points $\left(p, p\right), \left(-p, -p\right), \left(p\sqrt{3}, -p\sqrt{3}\right)$ are vertices of an equilateral triangle.

Show that the points $\left(4, 4\right), \left(5, -1\right), \left(-6, 2\right)$ are the vertices of a right-angled triangle.

Show that the points $A\left(4, 3\right), B\left(1, 2\right), C\left(1, 0\right)$ and $D\left(4, 1\right)$ are the vertices of a parallelogram and find the lengths of its diagonals.

Show that the points $A\left(2, 7\right), B\left(3, 0\right), C\left(-4, -1\right)$ are vertices of an isosceles triangle and find the length of base.

If the distance between $\left(-1, 3\right)$ and $\left(4, -2\right)$ is $k\sqrt{2}$. Then find the value of k.

If the distance between $\left(10, 4\right)$ and $\left(-1, -2\right)$ is $\sqrt{k}$. Then find the value of $k$.

If the distance between $\left(7, -2\right)$ and $\left(3, -1\right)$ is $\sqrt{k}$. Then find the value of k.

If the distance between $\left(1, 3\right)$ and $\left(7, 2\right)$ is $\sqrt{k}$, then $k=$

If the distance of the point $\left(-6, 12\right)$ from the origin is $k\sqrt{5}$. Then find the value of $k$.

If the distance of the point $\left(6, -10\right)$ from the origin is $2\sqrt{k}$, then find the value of $k$.

If the distance of the point $\left(2, 5\right)$ from the origin is $\sqrt{k}$. Then find the value of k.