Given the right triangle below, determine the distance of line segment AB. Round your final answer to the nearest tenth.

Steps to Solve

1. Identify the sides of the triangle In the right triangle, side $AC$ is the hypotenuse with a length of 14, side $BC$ is one leg with a length of 11.5, and side $AB$ is the other leg, which we need to find.

2. Apply the Pythagorean theorem The Pythagorean theorem states that for a right triangle: $$ AC^2 = AB^2 + BC^2 $$ Here, we will substitute the known values into the equation: $$ 14^2 = AB^2 + 11.5^2 $$

3. Calculate the squares of the known lengths Now we calculate the squares: $$ 14^2 = 196 $$ $$ 11.5^2 = 132.25 $$

4. Set up the equation to find $AB^2$ Substituting into the equation gives: $$ 196 = AB^2 + 132.25 $$

5. Subtract $132.25$ from both sides To isolate $AB^2$, subtract 132.25 from both sides: $$ AB^2 = 196 - 132.25 $$ $$ AB^2 = 63.75 $$

6. Take the square root of both sides Now, to find $AB$, take the square root: $$ AB = \sqrt{63.75} \approx 7.993 $$

7. Round to the nearest tenth Finally, we round to the nearest tenth: $$ AB \approx 8.0 $$