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 known values in the triangle

In the right triangle ABC, we know the lengths of AC and BC. AC = 14 and BC = 11.5. We will find the length of AB, which we denote as $x$.

2. Apply the Pythagorean theorem

The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides equals the square of the length of the hypotenuse. The equation is expressed as:

$$ c^2 = a^2 + b^2 $$

Here, $c$ is the hypotenuse (AC), and $a$ and $b$ are the other two sides (AB and BC).

3. Set up the equation using known values

Substituting the known values into the equation gives:

$$ 14^2 = x^2 + 11.5^2 $$

Calculating the squares:

$$ 196 = x^2 + 132.25 $$

4. Isolate $x^2$

Rearranging the equation to solve for $x^2$:

$$ x^2 = 196 - 132.25 $$

Calculating the right side:

$$ x^2 = 63.75 $$

5. Calculate $x$

To find $x$, take the square root of $x^2$:

$$ x = \sqrt{63.75} $$

Calculating the square root gives:

$$ x \approx 7.99 $$

6. Round to the nearest tenth

Finally, rounding $x$ to the nearest tenth:

$$ x \approx 8.0 $$