To find the angle between the side of the house and the cat's line of sight, we can visualize the scenario as a right triangle where:
- One leg of the triangle is the distance from the base of the house to the point directly below the cat (12 feet).
- The other leg is the height of the house, which we need to find.
- The hypotenuse is the diagonal distance from the cat to you, which is given as 18 feet.
Using the right triangle properties, we will denote the angle we want to find as \( \theta \), where \( \theta \) is the angle between the wall of the house (the opposite side) and the line of sight (the hypotenuse).
We can use the inverse sine function as follows:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\text{height of the house}}{18} \]
To find the height of the house, we can use the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{height}^2 + \text{base}^2 \]
Substituting the known values:
\[ 18^2 = \text{height}^2 + 12^2 \]
Calculating the squares:
\[ 324 = \text{height}^2 + 144 \]
Now, isolating the height:
\[ \text{height}^2 = 324 - 144 = 180 \]
Taking the square root:
\[ \text{height} = \sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5} \approx 13.42 \text{ feet} \]
Now we can find \( \theta \):
\[ \sin(\theta) = \frac{\text{height}}{18} = \frac{6\sqrt{5}}{18} = \frac{\sqrt{5}}{3} \]
Now we compute \( \theta \):
\[ \theta = \arcsin\left(\frac{\sqrt{5}}{3}\right) \]
Using a calculator to find the angle:
\[ \theta \approx 56.31^\circ \]
Rounding to the nearest whole degree:
\[ \theta \approx 56^\circ \]
Thus, the angle between the side of the house and the cat's line of sight looking down at you is approximately 56 degrees.