Question

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

Answer 1

`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`