Using the origin as one vertex, plot the vertices of:
   (A) A right-angled isosceles triangle.
   (B) An isosceles triangle with one vertex in Quadrant III and the other in Quadrant IV.

(A) A right-angled isosceles triangle with right angle at O and the two equal legs along the axes works well.
Consider:

$O = (0, 0)$ , $A = (4, 0)$ on the $x$ -axis, $B = (0, 4)$ on the $y$ -axis.

Here, $OA = OB = 4$ units (the two equal legs).
 Angle $AOB = 90^\circ$ (angle between the axes).

Hypotenuse $AB = \sqrt{16+16} = 4\sqrt{2}$ .

So, $\Delta OAB$ is right-angled and isosceles.

(B) To construct an isosceles triangle with O as one vertex, one vertex in Quadrant III ( $x < 0$ , $y < 0$ ) and another in Quadrant IV ( $x > 0$ , $y < 0$ ). Choose two points symmetric about the $y$ -axis so that their distances from O are equal.

Consider:

$O = (0, 0)$ , $P = (-3, -4)$ in Q-III, $Q = (3, -4)$ in Q-IV.

$$OP = \sqrt{9+16} = 5, OQ = \sqrt{9+16} = 5.$$

So, $OP = OQ$ , the triangle $OPQ$ is isosceles.