- Identify two possible cases for the third side of the right triangle: either both given sides are legs, or the longest side (15) is the hypotenuse.
- Compute the lengths using the Pythagorean theorem: \(c_1 = \sqrt{12^2+15^2} \approx 19.2\) and \(c_2 = \sqrt{15^2-12^2} = 9\).
- Find the difference between the two lengths: \(19.2 - 9 = 10.2\).
- The final answer is \(\boxed{10.2}\).
Explanation
- Analyze the scenarios for the side lengths.
We are given a right triangle with two sides of length 12 inches and 15 inches. In a right triangle, there are two distinct scenarios for how these two sides relate to the third side, depending on whether one of the given sides is the hypotenuse or if both are legs.
Case 1: Both given sides, 12 inches and 15 inches, are the legs of the right triangle. Let the third side be \(c_1\). By the Pythagorean theorem, \(c_1^2 = 12^2 + 15^2\).
Case 2: One of the given sides (15 inches) is the hypotenuse, and the other (12 inches) is a leg. Let the third side be \(c_2\). By the Pythagorean theorem, \(c_2^2 + 12^2 = 15^2\). (Note that 12 cannot be the hypotenuse, because the hypotenuse must be the longest side, and 15 is already given as a side.)
- Calculate the two possible lengths for the third side.
For Case 1, we calculate the hypotenuse \(c_1\):
\(c_1^2 = 12^2 + 15^2 = 144 + 225 = 369\)
\(c_1 = \sqrt{369} \approx 19.209\) inches.
For Case 2, we calculate the other leg \(c_2\):
\(c_2^2 = 15^2 - 12^2 = 225 - 144 = 81\)
\(c_2 = \sqrt{81} = 9\) inches.
- Find the difference and select the correct option.
Now we find the difference between the two calculated side lengths:
Difference \(D = |c_1 - c_2| \approx |19.209 - 9| = 10.209\).
Rounding the result to the nearest tenth, we get 10.2.
Comparing this with the given options:
The correct option is (d).
Examples
In construction or carpentry, identifying unknown side lengths of structural supports is a daily task. For instance, if you are building a triangular brace for a shelf and you know the dimensions of two sides, you must determine the third side correctly to ensure the frame is square and stable. If the design allows for two different, equally valid configurations for the brace, calculating the difference in their lengths helps the builder decide which one fits best within the available wall space. In this problem, we found that the difference between the two potential brace lengths is \(\boxed{10.2}\) inches.