Given information:
- A circle of diameter 8 inches is inscribed in triangle ABC.
- BC = 10 inches.

To find:
The area of the triangle ABC in square inches.

Approach:
To find the area of the triangle, we need the height and base of the triangle. In this case, the base is given as BC = 10 inches. We need to find the height of the triangle.

Solution:
1. Let O be the center of the inscribed circle and draw radii OA, OB, and OC to the points of tangency on the triangle.
2. Since the circle is inscribed in the triangle, the radii drawn from the center of the circle to the points of tangency are perpendicular to the sides of the triangle.
3. Therefore, OA, OB, and OC are perpendicular to AB, BC, and AC respectively.
4. Also, since OA, OB, and OC are radii of the circle, they are equal in length.
5. Let the length of OA (or OB or OC) be r.
6. Since OB is perpendicular to BC, the length of OB is the altitude (height) of the triangle from vertex B.
7. The triangle formed by OB, BC, and the line segment from O to the midpoint of BC is a right-angled triangle.
8. The length of BC is given as 10 inches, and the length of OB is r (radius of the circle).
9. By Pythagoras' theorem, we have:

OB² + (BC/2)² = OC²

r² + (10/2)² = (OC + r)²

r² + 5² = (OC + r)²

r² + 25 = OC² + 2OCr + r²

25 = OC² + 2OCr

OC = (25 - 2OCr)/OC

10. The length of OC can be expressed in terms of the radius r using the formula for the hypotenuse of a right-angled triangle.
11. Since OC is the sum of two sides of a right-angled triangle, which are r and r, we can use the Pythagorean theorem to find OC:

OC² = r² + r²

OC² = 2r²

OC = sqrt(2r²) = sqrt(2) * r

12. Substituting the value of OC in equation (9):

25 = (sqrt(2) * r)² + 2(sqrt(2) * r) * r

25 = 2r² + 2sqrt(2) * r²

25 = (2 + 2sqrt(2)) * r²

r² = 25 / (2 + 2sqrt(2))

13. The area of the triangle ABC can be calculated using the formula:

Area = (1/2) * base * height

14. The base is given as BC = 10 inches.
15. The height can be calculated using the radius r:

Height = OC + r = (sqrt(2) * r) + r

16. Substituting the value of r from equation (12) into equation (15):

Height = (