Q:

homework geometry question

I stand looking at a wall on which there are drawn three identical circles in a row. I hold up a coin with diameter 2cm at an arm’s distance (80cm) from my eyes (10cm apart). If I look through my left eye, the coin exactly covers the right circle, and if I look through my right eye, the coin exactly covers the left circle. How far do I stand from the wall?

A:

This is the view from above:

$A$ and $B$ are your eyes, $AB=10$
$GH=2$ is the coin
$EC=CD=DF=d$ are the 3 circles ($d$ is the unknown diameter)
distance from point $E$ to line $AB$ is $x$ (this is what you want to find)
distance from point $G$ to line $AB$ is 80 - your arm length
distance from point $J$ to line $AB$ is $a$ - unknown

From triangle similarity $ADF \sim AGH$: $$\frac{2}{80}=\frac{d}{x}$$
Thus $$x=40d$$
From triangle similarity $JGH \sim JAB$: $$\frac{2}{10}=\frac{a-80}{a}$$
Thus $$a=100$$
From triangle similarity $JGH \sim JCD$: $$\frac{2}{a-80}=\frac{d}{x-a}$$
Thus $$3d=10$$
and $$x=133\frac{1}{3}$$

