Question

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

Answer 1

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

Explanation:

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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