A point on the ceiling of a room is 12ft. from each of the vertices of a triangle on the floor whose legs are 6ft. A and 8ft. respectively. find the length of the ceiling.

To find the length of the ceiling, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, we have a right triangle on the floor with legs of lengths 6ft and 8ft, and a point on the ceiling that is 12ft from each vertex of the triangle.

Let's denote the length of the ceiling as c. The point on the ceiling forms two right triangles on the floor with the legs of the main triangle.

The first right triangle has one leg of length 6ft (from one vertex to the point on the ceiling) and another leg of length c (from the point on the ceiling to the other vertex of the main triangle). Applying the Pythagorean theorem, we have:

6^2 + c^2 = (12)^2

36 + c^2 = 144

c^2 = 144 - 36

c^2 = 108

Taking the square root of both sides, we get:

c = √108

Simplifying further, we have:

c = √(36 * 3)

c = 6√3

Therefore, the length of the ceiling is 6√3 ft.