How can I deduce the hypotenuse from the information given?

Question

I'm going into Machine Learning and am currently brushing up on some Calculus on Coursera. Everything was going smoothly until I got to this word problem:

A ladder rests against a wall. The top of the ladder touches the wall at height $12$ meters. The length of the ladder is $4$ meters longer than the distance from the base of the ladder to the wall. Find the length of the ladder.

I am confused as to how to deduce the hypotenuse from the information given above. And have sat here trying different things with no success. What am I missing?

Answer 1

Let $d$ be the distance from the ladder to the wall, and $l$ the length of the ladder. Then

$l = d + 4; \tag 1$

since the wall is mos' likely perpendicular to the ground, we may deploy the Pythagorean theorem and write

$l^2 = (12)^2 + d^2; \tag 2$

substituting (1) into (2) yields

$(d + 4)^2 = 144 + d^2, \tag 3$

$d^2 + 8d + 16 = 144 + d^2, \tag 4$

$8d + 16 = 144 \Longrightarrow 8d = 128 \Longrightarrow d = 16M \Longrightarrow l = 20M. \tag 5$

Answer 2

Given the length of the wall as $12$.

Take the length of the base as $x$.

Since the length of the ladder $l$ is $4$ meters greater than the base, we have $l = x+4$

Now according to the pythagorean theorem we have,

$\begin{align} (x+4)^2 &= 12^2 + x^2 \\ x^2 + 16 + 8x &= 144 + x^2 \\ 8x &= 128 \\ x & = 16 \end{align}$

So, the length of the ladder $l = x+4 = 16+4 = 20$