A triangle has corners at $(3 , 5 )$ , $(4 ,7 )$ , and $(4 ,6 )$ . What is the radius of the triangle's inscribed circle?

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Answer 1 · 2018-05-18T13:52:38

Radius of inscribed circle $color(brown )(r = 0.215$ units

$A(3,5), B(4,7), C(4,6)$

Using distance formula,

$bar(AB) = c = sqrt((3-4)^2+(5-7)^2) = sqrt5 = 2.236$

$bar(BC) = a = sqrt((4-4)^2 + (7-6)^2) = 1$

$bar(AC) = b = sqrt((3-4)^2 + (5-6)^2) = sqrt2 = 1.414$

Area of triangle knowing all three sides is given by

$A_t = sqrt(s (s-a)(s-b)(s-c))$

Where semi perimeter = $s = (a + b + c) / 2$

$s = (1 + 1.414 + 2.236) / 2 = 4.65/2 = 2.325$

$A_t = sqrt(2.325 (2.325-1) * (2.325-1.414) * (2.325 -2.236)) = 0.5$

Let r be the radius of incircle.

Then $r = A_t / s = 0.5 / 2.325 = color(brown)(0.215$ units

A triangle has corners at (3 , 5 )(3 , 5 ), (4 ,7 )(4 ,7 ), and (4 ,6 )(4 ,6 ). What is the radius of the triangle's inscribed circle?