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?

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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?

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Answer 1 · 2018-05-18T13:52:38

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

Explanation:

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

Using distance formula,

$¯ ¯¯¯¯ ¯ A B = c = \sqrt{{(3 - 4)}^{2} + {(5 - 7)}^{2}} = \sqrt{5} = 2.236$ $bar(AB) = c = sqrt((3-4)^2+(5-7)^2) = sqrt5 = 2.236$

$¯ ¯¯¯¯ ¯ B C = a = \sqrt{{(4 - 4)}^{2} + {(7 - 6)}^{2}} = 1$ $bar(BC) = a = sqrt((4-4)^2 + (7-6)^2) = 1$

$¯ ¯¯¯¯ ¯ A C = b = \sqrt{{(3 - 4)}^{2} + {(5 - 6)}^{2}} = \sqrt{2} = 1.414$ $bar(AC) = b = sqrt((3-4)^2 + (5-6)^2) = sqrt2 = 1.414$

Area of triangle knowing all three sides is given by

$A_{t} = \sqrt{s (s - a) (s - b) (s - c)}$ $A_t = sqrt(s (s-a)(s-b)(s-c))$

Where semi perimeter = $s = \frac{a + b + c}{2}$ $s = (a + b + c) / 2$

$s = \frac{1 + 1.414 + 2.236}{2} = \frac{4.65}{2} = 2.325$ $s = (1 + 1.414 + 2.236) / 2 = 4.65/2 = 2.325$

$A_{t} = \sqrt{2.325 (2.325 - 1) \cdot (2.325 - 1.414) \cdot (2.325 - 2.236)} = 0.5$ $A_t = sqrt(2.325 (2.325-1) * (2.325-1.414) * (2.325 -2.236)) = 0.5$

Let r be the radius of incircle.

Then $r = \frac{A_{t}}{s} = \frac{0.5}{2.325} = 0.215$ $r = A_t / s = 0.5 / 2.325 = color(brown)(0.215$ units

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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?

1 Answer

Explanation:

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

1 Answer

Explanation:

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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?