Let $ABCD$ be a square of side $2a$. Find the coordinates of the vertices of this square when $A$ coincides with the origin and $AB$ and $AD$ are along $OX$ and $OY$ respectively.

The base $PQ$ of two equilateral triangles $PQR$ and $PQR\text{'}$ with side $2a$ lies along $y-axis$ such that the mid-point of $PQ$ is at the origin. Find the coordinates of the vertices $R$ and $R\text{'}$ of the triangles.

Find the distance between the line segment joining the two points $(-\mathrm{6,7})$ and $(-1,-5)$.

Find the distance between the line segment joining the two points $\left(a+b, b+c\right) \mathrm{and} (a-b, c-b)$.

Find the distance between the line segment joining the two points $\left(a\sin \alpha , -b\cos \alpha \right) \text{and} (-a\cos \alpha , b\sin \alpha )$.

Find the distance between the points $(a,0)$ and $(0,b)$