Finding Isosceles Triangle Coordinates

In the coordinate plane, consider the vertices of triangle P(1, 2), Q(5, 6), and R(a, b). For the triangle to be isosceles with sides PQ and PR being equal, determine the valid coordinates for point R given that point Q lies at (5, 6). Additionally, the area of triangle PQR must equal 8 square units. What is one possible coordinate of point R?

Use the distance formula for equal lengths and the area formula for a triangle:

$$ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Area formula:

$$ \text{Area} = \f\frac{1}{2} \times | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| $$