The lengths of two sides of a right triangle are 12 inches and 15 inches. What is the difference between the two possible lengths of the third side of the triangle? Round your answer to the nearest tenth.

19.2

15

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

Let's assume that the third side is the hypotenuse. The given lengths of the two sides are 12 inches and 15 inches.

Using the Pythagorean theorem, we have:
c^2 = a^2 + b^2

where c is the hypotenuse and a and b are the other two sides.

Substituting the given values, we have:
c^2 = 12^2 + 15^2
c^2 = 144 + 225
c^2 = 369

Taking the square root of both sides, we find:
c ≈ √(369)
c ≈ 19.2

So one possible length of the third side is approximately 19.2 inches.

To find the other possible length, we can assume that the third side is one of the given sides.

If the third side is 12 inches, then the other side is the hypotenuse.

Using the Pythagorean theorem again, we have:
c^2 = a^2 + b^2
c^2 = 12^2 + 15^2
c^2 = 144 + 225
c^2 = 369

Taking the square root of both sides, we find:
c ≈ √(369)
c ≈ 19.2

Since 19.2 inches is already one of the possible lengths, the difference between the two possible lengths of the third side is 0 inches (or approximately 0 inches).

To find the difference between the two possible lengths of the third side of the right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Let's call the third side of the triangle "x", so we have two possible right triangles with side lengths (12, x) and (15, x). Applying the Pythagorean theorem, we can write the equations as follows:

For the first triangle:
12^2 + x^2 = hypotenuse^2

For the second triangle:
15^2 + x^2 = hypotenuse^2

Simplifying the equations:

144 + x^2 = hypotenuse^2
225 + x^2 = hypotenuse^2

Since the hypotenuse^2 is the same in both equations, we can equate the two equations:

144 + x^2 = 225 + x^2

Subtracting x^2 from both sides:

144 = 225

This is not possible, which means that there is no solution for the third side of the triangle. Therefore, the difference between the two possible lengths of the third side of the triangle is 0.

Rounding to the nearest tenth, the answer is 0.