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.