# A triangle has corners at #(2 , 5 )#, #(4 ,8 )#, and #(9 ,6 )#. What is the radius of the triangle's inscribed circle?

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See below.

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From diagram:

They also all have heights

So it follows:

Let:

Using the distance formula:

Now we need to find area of

There are different ways we can do this. Having the lengths of the sides we could use Heron's formula, or we could find the height of ABC using line equations. Since we have surds for the lengths of the sides, Heron's formula will be pretty messy and a calculator or computer will be really helpful.

Heron's formula is given as:

Where

I won't include the calculation for this, it's too big and too messy. I will just give the result.

So to find radius:

PLOT:

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