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### How Tall is Hal?

```
Date: 04/18/2001 at 13:12:50
From: Helen Narke
Subject: Problem

I have been out of school 30 years, and have no idea how to do this
problem:

Hal is standing 40 feet away from a 36-ft. tree. If the distance from
the top of the tree to the top of Hal's head is 50 ft., how tall is
Hal?

Thanks so much,
Helen
```

```
Date: 04/19/2001 at 13:48:44
From: Doctor TWE
Subject: Re: Problem

Hi Helen - thanks for writing to Dr. Math.

The best thing to do first (almost always) is to draw a picture of the
situation. Here's my diagram:

^
/ |
/   \|/
/     \ | /
/         \|/
/          \  |  /
/              \ | /
50' /               \  \|/  /
/                   \  |  /
/                       \ | /   36'
/                        \  \|/  /
/                            \  |  /
/                                \ | /
/                                    \|/
/________________________________________|
o                                         |
/|\                                        |
/ \_______________________________________/ \
Hal                  40'                  Tree

What we have here is a triangle on top of a rectangle. We want to find
the height of the rectangle. Let's simplify the drawing to "pure"
geometric shapes, eliminating the clutter:

^
/ |
/    |
/       |
/          |
/             |
/                |
50' /                   |
/                      |
/                         | 36'
/                            |
/                               |
/                                  |
/                                     |
/________________________________________|
|                                         |
|                                         |
|_________________________________________|
40'

We know that the top of the rectangle is 40', since it is the same
length as the bottom of the rectangle (opposites sides of rectangles
are equal in length). Thus, the base (bottom side) of the triangle is
40'.

Now one fact that you'd have to remember (or get out of a geometry
textbook) is called the Pythagorean theorem. It states:

The square of the hypotenuse of a right triangle is equal to the
sum of the squares of the other two sides.

It's also written algebraically as:

c^2 = a^2 + b^2

Where c is the hypotenuse, the longest side (the side going from the
top of Hal's head to the top of the tree).

Now we want to find the length of one of the sides of the triangle, so
we have to rearrange the Pythagorean theorem to solve for a.
Subtracting b^2 from both sides of the equation we get:

c^2 - b^2 = a^2
or
a^2 = c^2 - b^2

Plugging in our known values for c and b we get:

a^2 = (50)^2 - (40)^2
= 2500 - 1600
= 900

So what number squared gives us 900? We do the inverse of squaring --
we take the square root.

a = sqrt(a^2)
= sqrt(900)
= 30'

So the height of the tree *above Hal's head* is 30 feet. Subtracting
this from the total height of the tree, 36 feet, we see that Hal must
be 6 feet tall.

By the way, one "useful" triangle to know is the 3-4-5 triangle. If a
right triangle has sides of 3 and 4, the hypotenuse must be 5.
Likewise, if a right triangle has a side of 3 or 4 and a hypotenuse of
5, the other side must be 3 or 4 (whichever the first side was not).
Of course, this also applies to multiples of 3-4-5, like 6-8-10,
9-12-15, 12-16-20, 15-20-25, and so on. Knowing this, I could tell
that X-40-50 was a multiple of 3-4-5 (with each side multiplied by
10), so I could see that our 'a' side was 30 without having to do the
Pythagorean calculations. This can be a handy time saver on tests.

I hope this helps. If you have any more questions, write back.

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

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