Ladder measurement word problem

Question

a 15 ft. ladder is leaning against a wall. if the distance between the base of the ladder and the wall is 3 ft. less than the height of the ladder on the wall, how high up on the wall is the ladder?

Answer 1

So, This is a right triangle, with the ladder forming the hypotenuse (15 ft), and the distance between the wall and the base of ladder is the "Base". This is 3ft less in length than the height (Height or Altitude; H) on the wall where the ladder touches the wall. so, Base, B= (H - 3)ft

Pythagoras theorem: Hypotenuse^2 = Base^2 + Height^2

Here, therefore; 15^2 = (Height-3)^2 + Height^2

225 ft^2 = H^2 - 2Hx3 + 3^2 + Ht^2

225 = 2H^2 - 6H + 9

or, 2H^2 -6H +9 -225 = 0

Divinding by 2 the common factor and reducing the equation,

or, H^2  -3H - 108 = 0

Solving for H

H^2 +9H -12H - 108 =0

H(H+9) - 12(H+9) = 0

(H+9) (H - 12) = 0

So, either H +9 =0 or H-12=0

So, H is either  -9 or 12 ft

Height of the wall cannot be negative, so, the height where the ladder toches the wall is 12 ft high, and the Base, therefore is 12-3=9 ft.

Answer 2

the ladder, the height of the wall, and the horizontal distance from the wall to the ladder form a right triangle

the height of the wall and the horizontal distance are the legs and the ladder is the hypotenuse

15^2=n^2+(n-3)^2

225=n^2+n^2-6n+9

n^2+n^2-6n+9-225=0

2n^2-6n-216=0

n^2-3n-108=0

factor

108=12*9 and 12-9=3

(n-12)(n+9)=0

n-12=0

n=12 feet (answer)

n-3=9 feet (horizontal distance)