## Tuesday, May 29, 2012

### Poles and Strings

A pair of telephone poles d metres apart is supported by two cables which run from the top of each pole to the bottom of the other. The poles are 4 m and 6 m tall. Determine the height above the ground of the point, T , where the two cables intersect. What happens to this height as d increases? This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com
Let the height of the poles QP = a and RS = b.
Let the distance PO = c. Let h be the height above ground of the point T.

Since ∆QPR and ∆TOR are similar , so a : d = h : (d – c) , or (d – c ) = dh / a.
Similarly , ∆SRP and ∆TOP are similar , so b : d = h : c . So , c = dh / b.

Eliminating c from the above two equations we get d = dh (1/a + 1/b) , which gives h = ab / (a+b)
If a = 4 m and b = 6 m then we get h = 12 / 5 m. Thus height of point T is 12/5 m , independent of d.This can be seen by dragging points P or Q and see the path taken by point T.

1. This is absolutely correct. Just a slightly more elegant solution, using the same principle, might look like this:
Let PO=x, OR=y. So, d=x+y, but it's not important.
From similarity of one pair of triangles:
h/a=y/(x+y)
From similarity of another pair
h/b=x/(x+y)
Add these two together and you get:
h(1/a + 1/b) = (x+y)/(x+y)
That is,
h(a+b)/ab = 1
and
h = ab/(a+b)

2. Thank you Zor Shekhtman