Question 1190310
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Let's say we had these four points
A = (0,0)
B = (0,8)
C = (p,0)
D = (p,12)
I'll make p > 0, but it easily works for negative values of p as well.
The only condition really is that {{{p <> 0}}}


Joining points A and B gets segment AB which is the first tower (8 meters)


Joining points C and D gets segment CD which is the second tower (12 meters)


The distance between the base of each tower (A and C) is exactly p units. 
It will turn out that the value of p doesn't matter.


Let's find the equation of line BC
m = slope
m = (y2-y1)/(x2-x1)
m = (0-8)/(p-0)
m = -8/p
The y intercept of line BC is b = 8 because of point B on the y axis.
y = mx+b
y = (-8/p)x+8


The equation of line BC is: y = (-8/p)x+8


Through similar steps, the equation of line AD is y = (12/p)x


Let point E be the intersection of those two lines. The goal is to find the y coordinate of point E.


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Apply substitution to get
y = (-8/p)x+8
(12/p)x = (-8/p)x+8
12x = -8x + 8p
12x+8x = 8p
20x = 8p
x = 8p/20
x = 2p/5


Then plug this into either equation for BC or AD
y = (-8/p)*x + 8
y = (-8/p)*(2p/5) + 8
y = -16/5 + 8
y = -16/5 + 40/5
y = (-16 + 40)/5
y = 24/5
y = 4.8
Or
y = (12/p)*x
y = (12/p)*(2p/5)
y = 24/5
y = 4.8


Either way we get y = 4.8 as the height of the intersection of the two lines.


In either case, the p variables cancel out. 
This means that the horizontal distance between the poles doesn't matter.


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Answer: <font color=red>4.8 meters</font> (this value is exact).
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