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

Question

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

Sadie Allen · Accepted Answer

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] [Ans]

Sophia Alfred · Answer

undefined To find the radius of the triangle's inscribed circle, we can use the formula:

$$ r = \frac{2 \times \text{Area of the Triangle}}{\text{Perimeter of the Triangle}} $$

We first need to find the area and perimeter of the triangle. The perimeter can be found by summing the lengths of the three sides, and the area can be calculated using Heron's formula, given the lengths of the sides.

After finding the area and perimeter, we can substitute these values into the formula to find the radius of the inscribed circle.