# How can I deduce the hypotenuse from the information given?

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?

• I added the "algebra-precalculus" tag to your post. Cheers! – Robert Lewis Dec 10 '18 at 2:47
• Draw a picture with the ladder. Label the sides of the triangle formed by the ladder, the wall, and the floor. If you call the base x, the ladder is x+4. Then use the Pythagorean theorem. – Joel Pereira Dec 10 '18 at 2:47

## 2 Answers

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

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

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

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

• Oh my. I distributed the 2 exponent to d and 4 individually instead of multiplying the expression by itself. Not the first time this has gotten me. – Edward Severinsen Dec 10 '18 at 2:49
• @EdwardSeverinsen: we're all learners, my friend! – Robert Lewis Dec 10 '18 at 2:50
• No, the answer should be "at that angle, the ladder doesn't rest at the wall" ;-) – DonQuiKong Dec 10 '18 at 9:51
• @DonQuiKong I wouldn't get on it, that's for sure! – DoctorPenguin Dec 10 '18 at 11:48

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 $$l = x+4 = 16+4 = 20$$

• Nice graphic, +1! – Robert Lewis Dec 10 '18 at 2:52
• @RobertLewis Thanks! – Key Flex Dec 10 '18 at 2:57
• Just as a note 4 times greater than the base is ambiguous and could imply it is x * 4 and not x + 4. – Felix Guo Dec 10 '18 at 7:36
• @FelixGuo "4 times greater" isn't ambiguous at all: it can only mean $x\times 4$ and never $x+4$. It was just a mistake, which has now been corrected. – David Richerby Dec 10 '18 at 14:47