To solve the problem, we can use the Pythagorean theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case:
- The height the ladder needs to reach (i.e., the edge of the roof) is 24 feet (this is one leg of the triangle).
- The length of the ladder is the hypotenuse, which is 30 feet.
- We need to find the base of the ladder's distance from the wall of the house, which is the other leg of the triangle.
Let \( x \) be the distance from the base of the ladder to the wall. According to the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Where:
- \( a = 24 \) ft (the height of the roof),
- \( b = x \) (the distance from the wall), and
- \( c = 30 \) ft (the length of the ladder).
Plugging in what we know, we have:
\[ 24^2 + x^2 = 30^2 \]
Calculating the squares:
\[ 576 + x^2 = 900 \]
Next, we isolate \( x^2 \):
\[ x^2 = 900 - 576 \]
\[ x^2 = 324 \]
Now, take the square root of both sides to solve for \( x \):
\[ x = \sqrt{324} \]
\[ x = 18 \text{ ft} \]
Thus, the base of the ladder should be positioned 18 feet away from the house.
Summary:
The base of the ladder needs to be positioned 18.00 feet away from the house.
Diagram:
Here's a description of how to draw the right triangle based on the explanation above:
- Draw a vertical line representing the height of the roof (24 ft).
- Draw a horizontal line from the bottom of the vertical line to represent the ground (this is the distance x).
- Draw a diagonal line from the top of the vertical line to the end of the horizontal line representing the ladder (30 ft).
- Label the height as 24 ft, the base as 18 ft, and the hypotenuse as 30 ft.
Unfortunately, I'm unable to provide images directly, but you can easily create this diagram using any drawing tool or on paper based on this description.
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