A triangle has corners at #(2 , 4 )#, #(8 ,2 )#, and #(1 ,3 )#. What is the radius of the triangle's inscribed circle?

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Answer 1 · 2017-12-14T21:16:57

#0.5402#

Introduction: Law of Cotangents

To solve this question you will need to know about the #"Law of Cotangents"#.

Here is an article from Wikipedia about this law.

In brief, the states that the radius of a triangle's inscribed circle is

For a inscribed circle in a triangle with radius #r# and triangle with sides #a#, #b# and #c#:

#r = sqrt [ ((s - a)(s - b)(s - c)) / s ]# where #s = (a + b + c) / 2#

You can find the proof for this law in the Wikipedia article.

Step One: Find side #a#

#a = sqrt((8-2)^2+(2-4)^2) = sqrt(40)#

Step Two: Find side #b#

#a = sqrt((1-8)^2+(3-2)^2) = sqrt(50)#

Step Three: Find side #c#

#a = sqrt((2-1)^2+(4-3)^2) = sqrt(2)#

Step Four: Find the value of #s#

#s = (a + b + c) / 2 = (sqrt(40)+sqrt(50)+sqrt(2))/2 = sqrt(10) + 3sqrt(2)#

Step Five: Use the equation!

#r = sqrt [ ((s - a)(s - b)(s - c)) / s ] = 0.5402#

Assuming that you know how to use equations, I didn't put my working for using the equation.

Hope that makes sense!