There are two noncongruent triangles that have a 9-inch side, a 10-inch side, and that enclose 36 square inches of area. Find the length of the third side in each of these triangles.

To find the length of the third side in each of these triangles, we can use the formula for the area of a triangle. The formula is given by:

Area = (1/2) * base * height

Let x be the length of the third side in one of the triangles. Since we know the lengths of two sides (9 inches and 10 inches) and the area (36 square inches), we can set up an equation to solve for x.

The formula for the area of a triangle can be rearranged to solve for the height:

Height = (2 * Area) / base

Plugging in the values, we have:

Height = (2 * 36) / 10
Height = 72 / 10
Height = 7.2 inches

Now, to find the length of the third side, we can use the Pythagorean theorem. In a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Using the side lengths of 9 inches, 10 inches, and the height we found (7.2 inches), we can set up the equation:

x^2 = 9^2 - 7.2^2
x^2 = 81 - 51.84
x^2 = 29.16

Taking the square root of both sides, we find:

x = √29.16
x ≈ 5.4 inches

Therefore, the length of the third side in one of the triangles is approximately 5.4 inches. Since there are two noncongruent triangles with these properties, the length of the third side in the other triangle will be different.