a ladder is against a building. The distance from the bottom of the building to the ladder is 9 feet less than the length of the ladder. How high up the side of the building is the top of the ladder if that distance is 2ft. less than the length of the ladder?

Use the Pythagorean theorem and solve for x.

9^2 + (x-2)^2 = x^2

To find the height of the ladder, we can use the Pythagorean theorem. The ladder, the side of the building, and the distance from the bottom of the building to the ladder form a right triangle.

Let's visualize this problem:
1. Start by drawing a right triangle, where the vertical side represents the height of the building, the horizontal side represents the distance from the bottom of the building to the ladder, and the hypotenuse represents the ladder itself.

2. We are given that the distance from the bottom of the building to the ladder is 9 feet less than the length of the ladder. Let's call the length of the ladder "x". Therefore, the distance from the bottom of the building to the ladder is x - 9.

3. The problem also states that the height of the ladder is 2 feet less than its length. So, the height of the ladder is (x - 2).

Now, let's use the Pythagorean theorem to find the height of the building (the vertical side of the triangle):
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the equation becomes:
(x - 9)^2 + (x - 2)^2 = x^2

To solve this equation, we can expand and simplify it:

x^2 - 18x + 81 + x^2 - 4x + 4 = x^2
2x^2 - 22x + 85 = x^2

Subtracting x^2 from both sides:
x^2 - 22x + 85 = 0

We now have a quadratic equation that we can solve using factoring, completing the square, or the quadratic formula. By factoring or using the quadratic formula, we find that x = 17 or x = 5.

Since a ladder cannot be 5 feet long, we can discard the x = 5 solution. Thus, the length of the ladder is x = 17 feet.

To find the height of the building (the vertical side), we can substitute the value of x into our equation:

Height = x - 2
Height = 17 - 2
Height = 15 feet

Therefore, the height up the side of the building where the top of the ladder reaches is 15 feet.