Sylvia is replacing a piece of siding on her house. To make the 12 ft. ladder stable, the bottom of the ladder needs to be 6 ft. from the base of her house. Applying the Pythagorean Theorem and rounding to the nearest tenth, how high will the ladder reach so that Sylvia can replace the siding?(1 point) Responses 180 ft. 180 ft. 10.4 ft. 10.4 ft. 108 ft. 108 ft. 13.4 ft.

Question

GPT-4o mini · Answer

To find out how high the ladder will reach on the side of the house, we can use the Pythagorean Theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the ladder, in this case) is equal to the sum of the squares of the lengths of the other two sides.

We have:

The length of the ladder (hypotenuse) = 12 ft
The distance from the base of the house to the bottom of the ladder (one side) = 6 ft
The height the ladder reaches on the house (the other side) = h (which we need to find)

According to the Pythagorean Theorem:

\[ a^2 + b^2 = c^2 \]

where \( c \) is the length of the hypotenuse, one side \( a \) is the distance from the wall (6 ft), and the other side \( b \) is the height \( h \).

Plugging in the values:

\[ 6^2 + h^2 = 12^2 \]

Calculating the squares:

\[ 36 + h^2 = 144 \]

Now, isolate \( h^2 \):

\[ h^2 = 144 - 36 \] \[ h^2 = 108 \]

Now, take the square root of both sides to find \( h \):

\[ h = \sqrt{108} \] \[ h \approx 10.4 \text{ ft} \]

So, the ladder will reach approximately 10.4 ft up the side of the house. Therefore, the correct response is:

10.4 ft