# The top of a ladder leans against a house at a height of 12 feet. The length of the ladder is 8 feet more than the distance from the house to the base of the ladder. Find the length of the ladder?

Mar 19, 2016

$13 f t$

#### Explanation: The ladder leans against a house at a height $A C = 12 f t$
Suppose distance from the house to the base of ladder $C B = x f t$
Given is that length of ladder $A B = C B + 8 = \left(x + 8\right) f t$
From Pythagorean theorem we know that
$A {B}^{2} = A {C}^{2} + C {B}^{2}$, inserting various values
${\left(x + 8\right)}^{2} = {12}^{2} + {x}^{2}$
or $\cancel{{x}^{2}} + 16 x + 64 = 144 + \cancel{{x}^{2}}$
or $16 x = 144 - 64$
or $16 x = \frac{80}{16} = 5$
Therefore length of ladder $= 5 + 8 = 13 f t$

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Alternatively, one can assume length of ladder $A B = x f t$
This sets the distance from the house to the base of ladder $C B = \left(x - 8\right) f t$
Then proceed with setting up of equation under Pythagorean theorem and solve for $x$