A triangle has corners at #(4 , 5 )#, #(1 ,2 )#, and #(5 ,3 )#. What is the radius of the triangle's inscribed circle?

1 Answer
Harish Chandra Rajpoot
Jul 19, 2018

#0.849#

Explanation:

The area #\Delta# of triangle with vertices #(x_1, y_1)\equiv(4, 5)#, #(x_2, y_2)\equiv(1, 2)# & #(x_3, y_3)\equiv(5, 3)# is given by following formula

#\Delta=1/2|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|#

#=1/2|4(2-3)+1(3-5)+5(5-2)|#

#=4.5#

Now, the lengths of all three sides say #a, b# & #c# of given triangle are computed by using distance formula as follows

#a=\sqrt{(4-1)^2+(5-2)^2}=3\sqrt2#

#b=\sqrt{(4-5)^2+(5-3)^2}=\sqrt5#

#c=\sqrt{(1-5)^2+(2-3)^2}=\sqrt17#

hence, the semi-perimeter #s# of given triangle is computed as follows

#s=\frac{a+b+c}{2}#

#=\frac{3\sqrt2+\sqrt5+\sqrt17}{2}=5.3#

hence, the radius of inscribed circle is given as

#\frac{\Delta}{s}#

#=\frac{4.5}{5.3}#

#=0.849#

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