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

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Answer 1 · 2018-01-05T12:54:02

Radius of the triangle's inscribed circle is #1.07# unit.

The three corners are #A (1,2) B (5,2) and C (3,5)#

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 ((1-5)^2+(2-2)^2) =4 #unit

Side #BC= sqrt ((5-3)^2+(2-5)^2) ~~3.61#unit

Side #CA= sqrt ((3-1)^2+(5-2)^2) ~~ 3.61#unit

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

#s= (4+3.61+3.61)/2~~ 5.61# unit

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

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

#A_t = |1/2(-3+15+0)| = | 6.0| =6.0# sq.unit

Incircle radius is #r_i= A_t/s = 6.0/5.61 ~~1.07# unit

Radius of the triangle's inscribed circle is #1.07# unit [Ans]

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