We are asked to find the length of the dotted line in the given diagram. The diagram includes a rectangle with one side labeled as 3 and a right triangle attached to the bottom side of the rectangle, with hypotenuse 9 and one leg 5. We need to use the Pythagorean theorem twice to find the length of the dotted line.

1. Problem Description

We are asked to find the length of the dotted line in the given diagram. The diagram includes a rectangle with one side labeled as 3 and a right triangle attached to the bottom side of the rectangle, with hypotenuse 9 and one leg

5. We need to use the Pythagorean theorem twice to find the length of the dotted line.

2. Solution Steps

First, we will find the length of the horizontal side of the rectangle. This is the same as finding the length of the other leg of the right triangle with hypotenuse 9 and one leg

5. Let this length be $x$. Using the Pythagorean theorem, we have:

$x^2 + 5^2 = 9^2$

$x^2 + 25 = 81$

$x^2 = 81 - 25$

$x^2 = 56$

$x = \sqrt{56}$

Now, we can find the length of the dotted line. The dotted line is the hypotenuse of a right triangle with one leg of length 3 and the other leg of length $x = \sqrt{56}$.

Let the length of the dotted line be $d$. Using the Pythagorean theorem again, we have:

$d^2 = 3^2 + (\sqrt{56})^2$

$d^2 = 9 + 56$

$d^2 = 65$

$d = \sqrt{65}$

Now, we approximate $\sqrt{65}$ to the nearest tenth:

Since $8^2 = 64$, $\sqrt{65}$ is slightly larger than

8. $8.0^2 = 64$

$8.1^2 = 65.61$

Since $65$ is closer to $64$ than to $65.61$, we choose $8.1$ as our initial estimate.

We calculate $\sqrt{65} \approx 8.06$. Rounded to the nearest tenth, we have $8.1$.

3. Final Answer

8.1