Express the length of the as a function of x

Question

I am having problems understanding how to extract this information into a formula.

A 10-foot wall stands 5 feet from a building. A ladder, supported by the wall, touches a building. Express the length of the ladder as a function of x

(Note: i do have have the answer to the question, my question is on how to extract the information)

Any help would be much appreciated, Cheers!

Answer 1

By the Pythagorean Theorem, we get: $L=\sqrt{y^2+(x+5)^2}$.

Now, we have to find a way to express $y$ in terms of $x$.

Notice that the two triangles in the diagram are similar triangles. This means that the ratio of two sides in the first triangle has to be equal to the ratio of the corresponding sides in the second triangle:

$$\frac{x}{10}=\frac{x+5}{y}$$

By cross-multiplication, we find: $y=\frac{10(x+5)}{x}$.

Hence, we can express the length of $L$ by:

$$L=\sqrt{\bigg(\frac{10(x+5)}{x}\bigg)^2+(x+5)^2}$$

Answer 2

Which information are you trying to extract?

Assuming the question as given actually gave that x is the distance from the bottom of the ladder to the bottom of the wall, there are 4 pieces of information: 1) There is a 10 foot wall 2) The wall if 5 feet from the building 3) The ladder touches both the wall and the building 4) The distance from the bottom of the ladder to the bottom of the wall is x feet.

This is then enough information to produce the diagram given.

If you wish to solve the problem, I would then note that $$L^2 = y^2 + (x+5)^2$$ by pythagoras, And $$\frac{y}{5+x} = \frac{10}{x}$$ As they are similar triangles.