Aladder is leaning against a wall. the top of the ladder touches the wall at a height that is 5 feet less than the length of the ladder. the distance from the foot of the ladder to the wall is 10 feet less than the length of the ladder. how long is the ladder?

25 feet

Explanation:

Let the ladder be 'H' feet long

Given;

Ladder touches the wall at a height = H - 5 feet

The distance from the foot of the ladder to the wall = H - 10 feet

now a right angles triangle is formed by the system,

where,

Ladder forms the hypotenuse of the triangle

Height of the wall is the perpendicular

and, distance at the base is the base of the triangle formed

Therefore,

from the Pythagoras theorem, we have

H² = ( H - 10 )² + ( H - 5 )²

or

H² = H² + 100 - 20H + H² + 25 - 10H

or

H² = 2H² + 125 - 30H

or

H² -30H + 125 = 0

on solving the quadratic equation, we get

H² + ( - 5H - 25H ) + 125 = 0

or

H ( H - 5 ) - 25 (H - 5) = 0

or

(H - 5) × (H - 25) = 0

therefore,

we have

H = 5 feet and H = 25 feet

now,

H = 5 is not possible as this length of the ladder will lead to the negative distance at the base and also, the height of the at the wall be zero

Hence,

the length of the ladder is 25 feet

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