Question 1208824
<font color=black size=3>
Answer: <font color=red>2.875 meters</font>
This decimal value is exact and hasn't been rounded.


Explanation


For visual learners it often helps to draw a diagram if you're not sure where to start. 
{{{
drawing(400,400,-5,5,-5,5,
line(-2,-2,3,-2),line(3,-2,0,1),line(0,1,-2,-2),line(-2,-2,0.1814552942,-0.8325205567),line(0.1814552942,-0.8325205567,3,-2),line(0.1814552942,-0.8325205567,0,1),line(0.1814552942,-0.8325205567,0.1814552942,-2),line(0.1814552942,-0.8325205567,1.0069879254,-0.0069879254),line(1.0069879254,-0.0069879254,0.1814552942,-0.8325205567),line(0.1814552942,-0.8325205567,-0.7899463203,-0.1849194804),
circle(0.1814552942, -0.8325205567, 1.1674794433),
circle(-2,-2,0.08),circle(-2,-2,0.1),circle(-2,-2,0.12),circle(-2,-2,0.14),circle(3,-2,0.08),circle(3,-2,0.1),circle(3,-2,0.12),circle(3,-2,0.14),circle(0,1,0.08),circle(0,1,0.1),circle(0,1,0.12),circle(0,1,0.14),circle(0.1814552942,-0.8325205567,0.08),circle(0.1814552942,-0.8325205567,0.1),circle(0.1814552942,-0.8325205567,0.12),circle(0.1814552942,-0.8325205567,0.14),circle(0.1814552942,-2,0.08),circle(0.1814552942,-2,0.1),circle(0.1814552942,-2,0.12),circle(0.1814552942,-2,0.14),circle(1.0069879254,-0.0069879254,0.08),circle(1.0069879254,-0.0069879254,0.1),circle(1.0069879254,-0.0069879254,0.12),circle(1.0069879254,-0.0069879254,0.14),circle(1.0069879254,-0.0069879254,0.08),circle(1.0069879254,-0.0069879254,0.1),circle(1.0069879254,-0.0069879254,0.12),circle(1.0069879254,-0.0069879254,0.14),circle(-0.7899463203,-0.1849194804,0.08),circle(-0.7899463203,-0.1849194804,0.1),circle(-0.7899463203,-0.1849194804,0.12),circle(-0.7899463203,-0.1849194804,0.14),circle(-0.7899463203,-0.1849194804,0.08),circle(-0.7899463203,-0.1849194804,0.1),circle(-0.7899463203,-0.1849194804,0.12),circle(-0.7899463203,-0.1849194804,0.14),
locate(-2.2,-2.2,"A"),locate(2.8,-2.2,"B"),locate(-0.2,0.8+0.7,"C"),locate(-0.0185447058-0.15,-1.0325205567,"D"),locate(-0.0185447058,-2.2,"E"),locate(0.8069879254,-0.2069879254,"F"),locate(0.8069879254-2.3+0.2,-0.2069879254+0.2,"G")
)
}}}
We have triangle ABC with a circle inscribed inside it. This circle is known as an incircle.
This is the largest possible circle to fit inside the triangle where none of the circle spills outside it. Think of it like a balloon inflating to fill up a triangular enclosure.


The center of the incircle, known as the incenter, is point D.
Points E,F,G are tangent to the circle and reside on sides AB, BC, AC in that order.
Segments DE, DF, DG are radii of the incircle (these radius lengths are known as the inradius).
Since they are radii of the same circle, we know that <font color=blue>DE = DF = DG</font> which will be used later on below.


Since point E is one of the tangent points, it means inradius DE is perpendicular to segment AB. 
Similarly, DF is perpendicular to BC, and DG is perpendicular to CA.


Triangle ABC can be split into these three triangles
ABD, BCD, CAD


The base of triangle ABD is side AB. 
The height of triangle ABD is the inradius segment DE. 
Recall that the base and height are <u>always</u> perpendicular to each other.
area(triangle ABD) = 0.5*base*height = 0.5*AB*DE 


Similarly,
area(BCD) = 0.5*BC*DF
area(CAD) = 0.5*CA*DG


which leads to,
area(ABC) = area(ABD)+area(BCD)+area(CAD)
area(ABC) = 0.5*AB*DE + 0.5*BC*DF + 0.5*CA*DG
area(ABC) = 0.5*AB*<font color=blue>DE</font> + 0.5*BC*<font color=blue>DF</font> + 0.5*CA*<font color=blue>DG</font>
area(ABC) = 0.5*AB*<font color=blue>DE</font> + 0.5*BC*<font color=blue>DE</font> + 0.5*CA*<font color=blue>DE</font> ............  substitute DE in for the other radii
area(ABC) = 0.5*DE*(AB+BC+CA)
area(ABC) = 0.5*inRadius*(perimeter of triangle ABC)
We have a nice formula that connects the area, perimeter, and inradius of a triangle. 


Some textbooks will use this formula
triangleArea = inRadius*semiPerimeter
which is just a slight variation of the previous formula. The semi perimeter is half the perimeter.
i.e. semiperimeter = 0.5*perimeter


Let's solve for the inradius
area(ABC) = 0.5*inRadius*(perimeter of triangle ABC)
2*area(ABC) = inRadius*(perimeter of triangle ABC)
inRadius = 2*area(ABC)/(perimeter of triangle ABC)


Now use the info your teacher gave you.
inRadius = 2*area(ABC)/(perimeter of triangle ABC)
inRadius = 2*138/(96)
inRadius = <font color=red>2.875 meters exactly</font>. 
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