Two corners of an isosceles triangle are at #(5 ,2 )# and #(2 ,1 )#. If the triangle's area is #8 #, what are the lengths of the triangle's sides?

To find the lengths of the sides of the isosceles triangle, you can use the distance formula to calculate the distances between the given points. Then, use the formula for the area of a triangle, which involves the lengths of the sides.

Let A(5, 2) and B(2, 1) be the given coordinates of the triangle. The third vertex, let's call it C, is the apex of the isosceles triangle.

The distance between A and B (side AB) can be found using the distance formula.

The distance between A and C (side AC) and between B and C (side BC) will be equal since it's an isosceles triangle.

After finding the lengths of the sides, you can use Heron's formula or other methods to find the area of the triangle. Since the area is given as 8, you can set up the equation for the area and solve for the lengths of the sides.

This will give you the lengths of the sides of the isosceles triangle.