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```
Date: 05/22/2001 at 07:55:02
From: Andreza Rio Branco

How can I measure a tree using its and my shadows?
```

```
Date: 05/23/2001 at 08:21:25
From: Doctor Rick

Hi, Andreza, thanks for writing to Ask Dr. Math!

The secret is in the sun's rays: they fall on both you and the tree
from the same direction. We say the rays are parallel. Also, the tree
stands straight and so do you, so you and the tree are parallel.
parallel.

The top of the tree is joined to the shadow of the top of the tree by
a line that points back up to the sun. The top of your head and the
back up to the sun.

TREE
|\
| \
|  \
|   \
? |    \
|     \
|      \               YOU
|       \               |\
|        \            5'| \
__|_________\_____________|__\______
50'                  2.5'

When each side of one triangle is parallel to a side of another
triangle, the triangles are SIMILAR. When we use this word in everyday
life, we just mean "they are sort of the same, but not quite." When we
use it in math, it has a special meaning: "their shapes are exactly
the same, though their sizes don't have to be the same."

The "tree triangle" and the "you triangle" are similar.

The sides of two similar triangles have equal proportions. If the
horizontal side of one triangle is twice as long as the horizontal
side of the other triangle, then the vertical side of the first
triangle is twice as long as the vertical side of the other, and the
diagonal sides follow the same pattern.

These equal proportions let you figure out one length if you know
three others. In the picture above, I marked "your height" as 5 feet,

50   100
--- = --- = 20
2.5    5

The ratio is 20 to 1; that is, the tree's shadow is 20 times as long
as your shadow. Since the triangles are similar, the ratio of the
tree's height to your height is also 20 to 1. If your height is 5
feet, and the tree is 20 times as tall, then the tree's height is

20 * 5 feet = 100 feet

(The "*" is the multiplication sign.) This is how you can use shadow
lengths and your own height to measure the height of a tree!

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Measurement
Middle School Triangles and Other Polygons

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