To solve for the height at which the top of the ladder touches the house, we can use the Pythagorean theorem, which relates the lengths of the sides of a right triangle. In this case, the ladder forms the hypotenuse of the triangle, while the distance from the base of the ladder to the house and the height the ladder reaches on the house form the two legs of the triangle.
We have the following information:
- The length of the ladder (hypotenuse) \(c = 20\) feet
- The distance from the bottom of the ladder to the base of the house (one leg) \(a = 5\) feet
- The height at which the ladder touches the house (the other leg) \(b\), which we need to find.
According to the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Plugging in the known values:
\[ 5^2 + b^2 = 20^2 \]
Calculating \(5^2\) and \(20^2\):
\[ 25 + b^2 = 400 \]
Now, isolate \(b^2\):
\[ b^2 = 400 - 25 \] \[ b^2 = 375 \]
Next, take the square root of both sides to find \(b\):
\[ b = \sqrt{375} \]
To get a precise numerical value, we can simplify \(\sqrt{375}\):
\[ \sqrt{375} = \sqrt{25 \times 15} = \sqrt{25} \times \sqrt{15} = 5\sqrt{15} \]
Now, we can calculate the approximate value of \(b\):
\[ b \approx 5 \times 3.87298 \quad (\text{since } \sqrt{15} \approx 3.87298) \] \[ b \approx 19.3649 \]
Rounding to the nearest tenth, we have:
\[ b \approx 19.4 \]
Thus, the height above the ground at which the top of the ladder touches the house is approximately
\[ \boxed{19.4} \text{ feet.} \]
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