SOLUTION: There is a circle inside a triangle the bottom of the triangle is 240 kilometers, one side is 330 kilometers, and the other side is 270 kilometers. The circle is the largest possib

Question 57348: There is a circle inside a triangle the bottom of the triangle is 240 kilometers, one side is 330 kilometers, and the other side is 270 kilometers. The circle is the largest possible circle that can fit inside the triangle. What is the area in square kilometers, rounding to the nearest hundred.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Inscribed Circle
From this we see that the intersection of any two angle bisectors is the center if the inscribed circle. It follows that all three internal angle bisectors intersect at one point, which is the center of the inscribed circle, or "incircle". The perpendicular distance from point I to any of the sides is the radius, r, of the incircle. The area, K, of triangle ABC is the sum of the areas of AIB, BIC, and CIA. If we label the lengths of the sides of triangle ABC in traditional form, with side a opposite vertex A, b opposite B, and c opposite C, then the areas of these three small triangles are cr/2, ar/2, and br/2 because the height of each small triangle is the radius of the incircle. The sum of these areas is
K = (a+b+c)(r)/2, or
K = sr. Also,
r = K/s
where s = (a+b+c)/2 is the semiperimeter of triangle ABC.
-----------
I don't know what methods you have for solving your problem.
I found the above on-line at MathWorld Inscribed Circle.
You can find the area of your triangle (K) = (1/2)base*height
height=sqrt(330^2-120^2)
base = 240
So, K= 0.5*240*307.4=47187.21
s=(240+330+270)/2=420
Then r=K/s=112.35
Area of the circle =(pi)r^2
=(pi)(112.35)^2=39655.18 sq. km
Cheers,
Stan H.

RELATED QUESTIONS
if the hypotenuse of a right triangle is 37 kilometers and one of the legs is 35... (answered by ankor@dixie-net.com)

A circle is inscribed inside a right angled triangle with equal sides. What is the area... (answered by ikleyn)

There is a square with side length k. Inside the square is a single circle tangent to all (answered by greenestamps)

if there is an equilateral triangle of side 20 cm if 10 identical circle placed inside... (answered by ikleyn,rothauserc)

In my question there is a diagram which shows a circle with a circumference of 1cm being... (answered by MathLover1)

A side of an equilateral triangle is the diameter of a semi-circle. If the radius of the... (answered by ikleyn)

What is the radius of the largest circle that will fit inside a triangle that has two... (answered by ikleyn)

An equilateral triangle has a circle of radius 5.00 m inscribed inside it. The circle... (answered by dfrazzetto)

Find the area of the shaded part of this figure. Round the answer to one decimal place.... (answered by ankor@dixie-net.com)