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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,636

**Hi;**

**This was posted in another thread:**

**Two large towers are erected on a perfectly level piece of ground. We will call them tower A and tower B. Two marks are drawn on the towers near the top. When a line is drawn from the base of tower A to the mark on tower B it is 50 meters long. When a line is drawn from tower B to tower A's mark it is 40 meters long. The two lines intersect between the towers 10 meters above the ground. How far are the towers away from each other?**

**This a fairly thorny problem and there are nice solutions to it over in that thread. Let's see if we can solve it in Geogebra.**

**1) Put a point at (0,0). It will be called A.**

**2) Create a slider and set Min at 0 and Max at 40 with an increment of .01. Right click the slider and uncheck Fixed object.**

**3) Move the slider to 25 and create a point in the input bar of (a,0). It will be called B. Resize your screen until you see both points.**

**4) Use the circle with center and radius tool and click A and enter 50.**

**5) Make another one at B with a radius of 40.**

**6) Draw a perpendicular line to the x axis through B. This will represent the rightmost tower.**

**7) Use the intersection tool to find the points of intersection of the smaller circle and the y axis. D and C are then created. Hide C and the smaller circle.**

**8)Use the intersection tool to find the points of intersection of the larger circle and the perpendicular line. F and E are then created. Hide E and larger circle.**

**9) Use the move graphics tool and stretch out the x axis until the drawing looks well spaced.**

**10) Draw line segment BD and AF. You will see b =40 and f = 50 in the algebra pane.**

**11) Get the point of intersection of BD and AF, it will called G.**

**12) In the input bar enter y(G) to get just the y value of point G. It will be created in the Number column of the algebra pane labeled g.**

**13) Right click g and drag it to the drawing and place right under the slider so you can see it easily.**

**14) In options set rounding to 15 decimal places and move the slider until g is as close to 10 as possible. I got g = 10.00700381196065. That is not bad. Can we do better?**

**15) Right click your slider and set increment to .0001. I got g = 10.000117605127501.**

**16) Keep reducing the increment until you get as close to 10 as possible.**

**With the smallest increment of .00000001, I got g = 10.000000005695588. Now read the value in the slider, it is a = 37.355085329999866. That is the distance between the towers! It agrees well with the exact answer. You should have a drawing that looks something like the one below.**

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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