Let's work through the problem step-by-step.
### Part I: Sketch and label a figure that illustrates the scenario above
Consider a right triangle where:
- The ladder forms the hypotenuse of the triangle, measuring 12 feet.
- The base of the ladder, which is 5 feet away from the house, forms one leg of the triangle.
- The distance from the ground up to where the ladder touches the house forms the other leg of the triangle.
Label the right-angle triangle:
- Hypotenuse (ladder length): 12 feet
- One leg (base distance from the house): 5 feet
- Other leg (distance up the house, marked as [tex]\( b \)[/tex])
### Part II: Set up an equation to find how far up the house the ladder reaches, and solve the equation. Round your answer to the nearest tenth.
We can use the Pythagorean theorem to solve for [tex]\( b \)[/tex], the height the ladder reaches up the house. The Pythagorean theorem states:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the legs of the right triangle, and [tex]\( c \)[/tex] is the hypotenuse.
In this problem:
- [tex]\( a = 5 \)[/tex] feet (base distance from the house).
- [tex]\( c = 12 \)[/tex] feet (ladder length).
Let's substitute these values into the Pythagorean theorem:
[tex]\[ 5^2 + b^2 = 12^2 \][/tex]
Now, solve for [tex]\( b \)[/tex]:
[tex]\[ 25 + b^2 = 144 \][/tex]
Subtract 25 from both sides:
[tex]\[ b^2 = 144 - 25 \][/tex]
[tex]\[ b^2 = 119 \][/tex]
To find [tex]\( b \)[/tex], take the square root of 119:
[tex]\[ b = \sqrt{119} \][/tex]
Using the calculated value, we find:
[tex]\[ b \approx 10.908712114635714 \][/tex]
Rounded to the nearest tenth:
[tex]\[ b \approx 10.9 \][/tex]
Therefore, the ladder reaches approximately 10.9 feet up the house.
So, the distance up the house that the ladder reaches is roughly 10.9 feet.
