A ladder is leaning against a building so that the distance from the ground to the top of the ladder is 4 feet less than the length of the ladder. Find the length of the ladder if the distance from the bottom of the ladder to the building is 8 feet.

Use Pythagoras.

Let me know what you find and how.

Let's call the length of the ladder "L". According to the problem, the distance from the ground to the top of the ladder is 4 feet less than the length of the ladder.

So, the height of the ladder on the building is L - 4.

The distance from the bottom of the ladder to the building is given as 8 feet.

Now, we can form a right triangle using the ladder as the hypotenuse, the height of the ladder on the building as one leg, and the distance from the bottom of the ladder to the building as the other leg.

Using the Pythagorean theorem, we can write the equation:

(L - 4)^2 = 8^2

Expanding and simplifying the equation, we get:

L^2 - 8L + 16 = 64

Subtracting 64 from both sides, we have:

L^2 - 8L - 48 = 0

Now, we can factor the quadratic equation:

(L - 12)(L + 4) = 0

Setting each factor equal to zero, we get:

L - 12 = 0 or L + 4 = 0

Solving these equations, we find:

L = 12 or L = -4

Since the length of the ladder cannot be negative, we can conclude that the length of the ladder is 12 feet.

To find the length of the ladder, we can create a right triangle with the ladder as the hypotenuse, and the distance from the ground to the top of the ladder as the height of the triangle. The distance from the bottom of the ladder to the building forms the base of the triangle.

Let's use the Pythagorean theorem to solve for the length of the ladder. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b):

c^2 = a^2 + b^2

In this case, we are given that the distance from the bottom of the ladder to the building is 8 feet (b), and the distance from the ground to the top of the ladder is 4 feet less than the length of the ladder (a = c - 4).

Plugging these values into the Pythagorean theorem equation, we get:

c^2 = (c - 4)^2 + 8^2

Now, let's solve this equation step by step:

Expand (c - 4)^2:

c^2 = c^2 - 8c + 16 + 64

Combine like terms:

0 = -8c + 80

Move -8c to the left side:

8c = 80

Divide by 8:

c = 10

Therefore, the length of the ladder is 10 feet.

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