Question

A ladder is leaning against a building so that the distance from the ground to the top of the latter is 1 foot less than the length of the latter find the length of the latter is the distance from the bottom of the ladder to the building is 7 feet

Answer 1

`25\text{ feet}`

Explanation:

The ladder, ground, and building form a right triangle where the vertical distance between the top of the ladder and the bottom of the ground is one leg, the horizontal distance between the bottom of the ladder and the building is another leg, and the length of the ladder is the hypotenuse.

For any right triangle, the Pythagorean Theorem states that the sum of the squares of both legs is equal to the hypotenuse squared (
`a^2+b^2=c^2`), where
`c` is the hypotenuse.

Let the length of the ladder be
`\ell` (hypotenuse of right triangle). The distance between the top of the ladder and the ground (vertical distance) can be represented as
`\ell -1`.

From the Pythagorean Theorem, we then have:

`7^2+(\ell-1)^2=\ell^2`

Expand using
`(a-b)^2=a^2-2ab+b^2`:

`49+\ell^2-2\ell+1=\ell^2`

Subtract
`\ell` from both sides and add
`2\ell` to both sides:

`49+1=2\ell`

Combine like terms:

`50=2\ell, \\2\ell =50`

Divide both sides by 2:

`\ell=(50)/(2)=\boxed{25\text{ feet}}`

Therefore, the length of the ladder is 25 feet.