A triangle has corners at #(3 , 5 )#, #(4 ,2 )#, and #(8 ,4 )#. What is the radius of the triangle's inscribed circle?

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Answer 1 · 2018-02-08T11:45:54

Radius of triangle's inscribed circle is #1.1# unit.

The three corners are #A (3,5) B (4,2) and C (8,4)#

Distance between two points #(x_1,y_1) and (x_2,y_2)# is

#D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2#

Side #AB= sqrt ((3-4)^2+(5-2)^2) ~~ 3.16#unit

Side #BC= sqrt ((4-8)^2+(2-4)^2) ~~4.47#unit

Side #CA= sqrt ((8-3)^2+(4-5)^2) ~~ 5.1 #unit

The semi perimeter of triangle is #s=(AB+BC+CA)/2# or

#s= (3.16+4.47+5.1)/2~~ 12.73/2~~ 6.37# unit.

Area of Triangle is #A_t = |1/2(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))|#

#A_t = |1/2(3(2−4)+4(4−5)+8(5−2))|# or

#A_t = |1/2(-6-4+24)| = | 7.0| =7.0# sq.unit.

Incircle radius is #r_i= A_t/s = 7.0/6.37 ~~1.1# unit

Radius of triangle's inscribed circle is #1.1# unit[Ans]