How high is the ladder up the side of the wall

Question

A ladder is leaning against a building. The distance from the bottom of the ladder to the building is 16ft shorter than the length of the ladder. How high up is the side of the building is the top of the ladder if that distance is 8ft less than the length of the ladder

Answer 1

The building forms a right angle with the ground, so we have a right triangle formed by the building, the ground, and the ladder.

 

let x = length of the ladder; then x - 16 = the length from the base of the ladder to the building, and x - 8 = height of the building

 

we will use the Pythagorean theorem that c² = a² + b² where c = hypotenuse of a right triangle

 

in this case, x² = (x - 8)² + (x - 16)²

expanding, we get x² = x² -16x + 64 + x² - 32x + 256

combining, we get 0 = x² - 48x + 320

this expression can be factored into (x - 8)(x- 40)

so, our roots are 8 and 40

 

but 8 will not function as a root, so it must be discarded, 

therefore x = 40; so, x - 8 = 32, and x - 16 = 24

 

∴ the length of the ladder is 40, the height of the building is 32, the length of the base is 24

 

Proof: 40² = 32² + 24²; 1600 = 1024 + 576; 1600 = 1600