Distance to an Object

Date: 04/07/2000 at 01:50:17
From: Brent Pattison
Subject: Easy way to calculate distance using trigonometry

I'm a middle school teacher who wants to give his students some 
hands-on application work in geometry. One idea I had was to get 
students to measure the distance from one object to another in our 
schoolyard using geometry. I remember that if one knows the 
measurement of a base line and both angles leading towards the object, 
the distance can be calculated. Is there an easy way I could show my 
students to do this, without having to teach them all about 
trigonometry? I think they'd get a kick out of watching math work this 
way!

Thanks.

Date: 04/07/2000 at 13:14:16
From: Doctor Rob
Subject: Re: Easy way to calculate distance using trigonometry

Thanks for writing to Ask Dr. Math, Brent.

You can use similar triangles to do this, instead of trigonometry. Use 
the baseline and the two angles:

         C
         o._
        /.  `-._a
      b/ .d     `-._
      /  .          `-._
     o---+--------------`o
    A    X    c           B

Let AB be the baseline with length c. Let the object be C, and let 
the two angles be A = <BAC and B = <ABC. You want to find the lengths 
a and b. 

Construct a small triangle with the same shape as ABC, that is, a 
similar triangle. You can do this because you know angles A and B (and 
thus C = 180 - A - B degrees). Measure its sides a', b', and c', and 
use the known value of c to find a = a'*(c/c') and b = b'*(c/c').

If you want to find d, the altitude of the triangle, you can do that 
by using Hero's or Heron's Formula to get the area K of triangle ABC,

     K = sqrt(s*[s-a]*[s-b]*[s-c]),

where  s = (a+b+c)/2, and then d = 2*K/c.

Not only does this use similar triangles, it uses A+B+C = 180 degrees, 
Hero's formula, and K = base*height/2. You can also talk about the 
accuracy of the result and significant figures, based on the accuracy 
of the angle and length measurements.

This kind of mathematics goes back to the Egyptians, who invented 
surveying to reestablish the boundaries of fields flooded during the 
annual floods of the Nile. You can even bring in the history of 
mathematics in this way.

Sounds like fun to me!

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/