<PRE><B>Question 838958</B>
a picture of your situation is shown below:
&lt;img src = &quot;http://theo.x10hosting.com/2014/feb083.jpg&quot; alt = &quot;$$$$&quot; &lt; /&gt;
EB is the fence post that is 6.5 feet high.
marta is 4 feet behind it (the length of ED).
marta eye level is 5 feet high (the length of DC).
the building is 32.5 feet from the fence (the length of FE).
I extended some of the measurements as follows:
GC is parallel to FD and 5 feet from it.
IB is parallel to GH and 1.5 feet from it.
the fence represented by EB was subdivided into 2 parts.
EH part is 5 feet and HB part is 1.5 feet.
so you have 2 rectangles of interest formed and 2 triangles of interest formed.
the rectangles of interest are:
GCDF
IBHG
the triangles of interest are:
BCH
ABI
the angles marked x are both equal to each other because they are corresponding angles of parallel lines.
the parallel lines are GC and IB
the angles marked y are both equal to each other because they are complementary angles to angles that are equal to each other.
angle y in triangle BCH is complementary to angle x in triangle BCH.
angle y in triangle ABI is complementary to angle x in triangle ABI.
this makes triangle ABI similar to triangle BCH by AA.
since corresponding sides of similar triangles are proportional, this means that AI is to BH as IB is to HC
this ratio can be shown as:
AI / BH = IB / HC
substituting their values in that equation, we get:
x / 1.5 = 32.5 / 4
the x being referred to here is the length of line segment AI.
cross multiply to get:
4 * x = 1.5 * 32.5
divide both sides of this equation by 4 to get:
x = (1.5 * 32.5) / 4
simplify to get:
x = 12.1875
the height of the building will be:
12.1875 + 1.5 + 5 = 18.6875 which is equal to 18.5 feet rounded to the nearest half a foot
there is another triangle formed that we can also use to verify with.
that triangle is triangle ACG.
the height of that triangle is equal to 12.1875 + 1.5 which is equal to 13.6875.
the base of that triangle is equal to 32.5 + 4 which is equal to 36.5.
that triangle is also similar to triangle BCH and the ratios are:
13.6875 / 36.5 = 1.5 / 4
cross multiply to get:
13.6875 * 4 = 36.5 * 1.5
simplify to get:
54.75 = 54.75
this confirms again that the solution is good.
the height of the building is:
12.1875 + 1.5 + 5 which is equal to 18.6875 which rounds to 18.5 feet.

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