Final answer:
The third vertex of an isosceles triangle placed on the coordinate plane with its base along the x-axis should have an x-coordinate that is the midpoint of the base vertices' x-coordinates, and its y-coordinate should maintain vertical symmetry. None of the given options are correct for the third vertex's coordinates; the correct coordinates would be (a, b).
Step-by-step explanation:
When Haresh is placing an isosceles triangle on the coordinate plane with its base along the x-axis, the third vertex of the triangle (which is not on the x-axis) should be placed such that it is equidistant from the two base vertices regarding the x-coordinate and at the same vertical level (same y-coordinate) as those two vertices to maintain symmetry. Therefore, the x-coordinate of the third vertex should be halfway between the x-coordinates of the base vertices, and the y-coordinate should be the same for both the base vertices to make sure the lengths of the sides of the triangle (other than the base) are equal. If we assume the base vertices are at coordinates (0,0) and (2a,0), the third vertex should be at (a,b).
None of the options provided, (b, 2b), (2a, b), (a/2, b), (a², b), (a², b²), are correct because they either do not align with the symmetry (equal distance from base vertices) of an isosceles triangle or do not maintain the same y-coordinate for the base vertices. However, option 3) (a/2, b) closely resembles what the third vertex's coordinates could be if the base were located at (0,0) and (a,0), but Haresh's triangle has points (0,0) and (2a,0), so the midpoint x-coordinate should be a, not a/2.