To find out how far up the wall the ladder reaches, we will use the Pythagorean theorem. This theorem applies to right-angled triangles and states that 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 scenario, the ladder serves as the hypotenuse of a right-angled triangle, with the wall and the ground forming the other two sides of the triangle. The distance from the foot of the ladder to the wall is one side of the triangle, and the height the ladder reaches up the wall is the other side.
Let's denote the following:
- \( c \) is the length of the ladder (hypotenuse),
- \( a \) is the distance from the foot of the ladder to the wall (one side),
- \( b \) is the height the ladder reaches up the wall (the other side).
From the Pythagorean theorem, we have:
\[ c^2 = a^2 + b^2 \]
Given that:
- The ladder is \( c = 10 \) m long (the hypotenuse),
- The distance from the ladder's foot to the wall is \( a = 6 \) m,
we want to find \( b \), the height up the wall.
Rearranging the equation to solve for \( b^2 \), we get:
\[ b^2 = c^2 - a^2 \]
We plug in the given values:
\[ b^2 = 10^2 - 6^2 \]
\[ b^2 = 100 - 36 \]
\[ b^2 = 64 \]
To find \( b \), we take the square root of both sides:
\[ b = \sqrt{64} \]
\[ b = 8 \]
Hence, the ladder reaches 8 meters up the wall.
