# A triangle has corners at #(3 , 2 )#, #(1 ,7 )#, and #(5 ,4 )#. What is the radius of the triangle's inscribed circle?

The radius of the incircle is

The length of the sides of the triangle are

The area of the triangle is

The radius of the incircle is

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To find the radius of the inscribed circle of a triangle, you can use the formula:

[ r = \frac{{\text{Area of the triangle}}}{{\text{Semi-perimeter of the triangle}}} ]

First, find the lengths of the sides of the triangle using the distance formula:

[ \text{Side 1} = \sqrt{(3 - 1)^2 + (2 - 7)^2} ] [ \text{Side 2} = \sqrt{(3 - 5)^2 + (2 - 4)^2} ] [ \text{Side 3} = \sqrt{(1 - 5)^2 + (7 - 4)^2} ]

Then, calculate the semi-perimeter of the triangle:

[ s = \frac{{\text{Side 1} + \text{Side 2} + \text{Side 3}}}{2} ]

Next, compute the area of the triangle using Heron's formula:

[ \text{Area} = \sqrt{s(s - \text{Side 1})(s - \text{Side 2})(s - \text{Side 3})} ]

Finally, plug the area and semi-perimeter values into the formula for the radius:

[ r = \frac{{\text{Area}}}{{s}} ]

This will give you the radius of the inscribed circle of the triangle.

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