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/