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

Understand the Problem

The question is asking us to calculate the length of line segment AB in a right triangle using the Pythagorean theorem. We know two sides of the triangle: AC is 14 units and BC is 11.5 units. We need to find the length of AB and round the final answer to the nearest tenth.

Answer

$18.1$

Steps to Solve

Identify the sides of the triangle

In the right triangle ( ABC ), we know that:

( AC = 14 ) units (one leg),
( BC = 11.5 ) units (the other leg),
( AB ) is the hypotenuse (the side we want to find).

Apply the Pythagorean theorem

The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. We can express this as:

$$ AB^2 = AC^2 + BC^2 $$

Substitute the known values into the equation

Replace ( AC ) and ( BC ) with their respective lengths:

$$ AB^2 = 14^2 + 11.5^2 $$

Calculating the squares:

$$ AB^2 = 196 + 132.25 $$

Simplify and calculate ( AB^2 )

Now, add the two results:

$$ AB^2 = 196 + 132.25 = 328.25 $$

Calculate ( AB ) by taking the square root

Now, to find ( AB ), we take the square root:

$$ AB = \sqrt{328.25} $$

Using a calculator gives:

$$ AB \approx 18.1 $$ (rounded to the nearest tenth)

The length of line segment ( AB ) is approximately $18.1$ units.

More Information

The Pythagorean theorem is a fundamental principle used in various fields, including physics, engineering, and architecture. The theorem can help determine distances and measurements when dealing with right triangles.

Tips

Forgetting to square both legs before adding them together.
Not using the square root to find the length of the hypotenuse.
Rounding off too early in the calculations instead of at the final step.

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