In a right triangle, $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse. Given $a = 3$ millimeters and $c = 6$ millimeters, find the length of $b$, rounded to the nearest tenth.

1. Problem Description

In a right triangle, $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse. Given $a = 3$ millimeters and $c = 6$ millimeters, find the length of $b$, rounded to the nearest tenth.

2. Solution Steps

We will use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a$ and $b$).

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

We are given $a = 3$ and $c = 6$, and we want to find $b$. Substituting the given values into the Pythagorean theorem gives:

$3^2 + b^2 = 6^2$

$9 + b^2 = 36$

Subtract 9 from both sides:

$b^2 = 36 - 9$

$b^2 = 27$

Take the square root of both sides:

$b = \sqrt{27}$

$b \approx 5.196$

Round to the nearest tenth:

$b \approx 5.2$

3. Final Answer

$b = 5.2$ millimeters