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/
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