The problem asks for the length of the dotted line in the diagram, rounded to the nearest tenth. The diagram contains a rectangle with a height of 3 and a right triangle with hypotenuse 9 and one leg of length 5. The dotted line is the hypotenuse of a right triangle, where one leg is the height of the rectangle (3) and the other leg is the same length as the other leg of the lower right triangle.

1. Problem Description

The problem asks for the length of the dotted line in the diagram, rounded to the nearest tenth. The diagram contains a rectangle with a height of 3 and a right triangle with hypotenuse 9 and one leg of length

5. The dotted line is the hypotenuse of a right triangle, where one leg is the height of the rectangle (3) and the other leg is the same length as the other leg of the lower right triangle.

2. Solution Steps

First, let's find the length of the horizontal leg of the lower right triangle using the Pythagorean theorem. Let's call the unknown length $x$. We have $x^2 + 5^2 = 9^2$.

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

$$x^2 + 25 = 81$$

$$x^2 = 81 - 25$$

$$x^2 = 56$$

$$x = \sqrt{56}$$

Next, we can find the length of the dotted line. Let's call it $d$. The dotted line is the hypotenuse of a right triangle with legs of length 3 and $\sqrt{56}$. Using the Pythagorean theorem, we have $d^2 = 3^2 + (\sqrt{56})^2$.

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

$$d^2 = 9 + 56$$

$$d^2 = 65$$

$$d = \sqrt{65}$$

Now, we need to approximate $\sqrt{65}$ to the nearest tenth. Since $8^2 = 64$, $\sqrt{65}$ will be slightly greater than

8. We can estimate $\sqrt{65} \approx 8.1$. To verify, $8.1^2 = 65.61$, which is close to

6

5. A more accurate estimation is $8.06$.

Therefore, $d = \sqrt{65} \approx 8.1$.

3. Final Answer

8.1