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A surveyor needs to determine the height of a tree. She places a mirror on the ground and paces backwards until she sees the top of the tree reflected in the mirror. She marks where she is standing and then measures the distance from this spot to the mirror which is $10$ feet. She then measures the distance from the mirror to the base of the tree which is $90$ feet. The surveyor's height is $54$ inches from the ground to her eyes. How tall is the tree (in feet) assuming level ground from the surveyor's spot to the mirror and from the mirror to the tree?

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You can use similarities to solve this. –  Sigur Feb 5 '13 at 0:17
    
What have you tried? I humbly recommend drawing your own picture (even if there's already one in the book). –  machine yearning Feb 5 '13 at 0:21
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How precisely horizontally can you get the mirror to lie? Doesn't sound like a very reliable method. A bowl of mercury might be slightly better (if you ignore health and environmental risks), but why doesn't the surveyor just drive home and fetch her theodolite already? –  Henning Makholm Feb 5 '13 at 0:27

1 Answer 1

Let $T$ be the top of the tree, $R$ the base of the tree, $M$ the mirror, $F$ the point where she’s standing, and $E$ her eye. The angle of incidence equals the angle of reflection, so the triangles $\triangle TRM$ and $\triangle EFM$ are similar. This implies that if the height of the tree is $h$ inches, then

$$\frac{h}{54}=\frac{90}{10}\;.$$

Now just do a little algebra to get $h$, and then convert to feet.

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