A triangle has corners at `(4 ,1 )`, `(2 ,6 )`, and `(7 ,3 )`. What is the radius of the triangle's inscribed circle?

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A triangle has vertices at `(4,1), (2,6) and (7,3)`.

Plot the points on a Cartesian Coordinate Plane and label them as `A, B and C` respectively.

Measure the magnitudes of the line segments `bar (AB), bar (BC) and bar (AC)`

Perimeter of the triangle `AB + BC + AC`

`rArr 5.39 + 5.83 + 3.61`

`rArr 14.83` Units

Semi-Perimeter [ s ] `=(Perimeter)/2`

`rArr 14.83/2` Units

`rArr 7.415` Units

Next, construct angle bisectors..

These three angle bisectors intersect at a point called Incenter.

Using the Incenter as one point and the three sides `AB, BC and AC`, construct perpendicular lines and mark the points where they intersect the sides of the triangle.

Measure the length of these lines from the Incenter

Construct a Circle, the center being the Incenter and one of the points on the sides as the Radius.

Note that all of them have the same magnitude `1.28` units.

We can also use the formula given below to find the magnitude of the radius.

`r^2 = [ (s-a)(s-b)(s-c) ]/s`

`rArr r^2=[(7.415-5.83)(7.415-3.61)(7.415-5.39)]/7.415`

`rArr r^2 = 12.21262313/7.415`

`rArr r^2 = 1.647015931`

`rArr r = sqrt(1.647015931)`

`rArr r ~~ 1.283361185`

`r~~1.28` Units.

Hence, Radius of the Inscribed circle `~~1.28` units.

Hope it helps.