To find out how high above the ground the top of the ladder touches the house, we can use the Pythagorean theorem. The ladder, the distance from the base of the house to the bottom of the ladder, and the height at which the ladder touches the wall form a right triangle.
Let:
- \( c \) be the length of the ladder (the hypotenuse), which is 20 feet.
- \( a \) be the distance from the bottom of the ladder to the base of the house (one leg of the triangle), which is 5 feet.
- \( b \) be the height at which the ladder touches the house (the other leg of the triangle), which we need to find.
According to the Pythagorean theorem, we know that:
\[ a^2 + b^2 = c^2 \]
Substituting the known values into the equation:
\[ 5^2 + b^2 = 20^2 \]
Calculating the squares:
\[ 25 + b^2 = 400 \]
To isolate \( b^2 \), subtract 25 from both sides:
\[ b^2 = 400 - 25 \]
\[ b^2 = 375 \]
Now, take the square root of both sides to solve for \( b \):
\[ b = \sqrt{375} \]
Calculating \( \sqrt{375} \):
\[ b \approx 19.3649 \]
Rounding this to the nearest tenth, we have:
\[ b \approx 19.4 \text{ feet} \]
Thus, the height above the ground where the top of the ladder is touching the house is approximately \( \boxed{19.4} \) feet.
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