A triangle has corners at $(3, 5)$ $(3 , 5 )$ , $(4, 2)$ $(4 ,2 )$ , and $(8, 4)$ $(8 ,4 )$ . What is the radius of the triangle's inscribed circle?

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A triangle has corners at $(3, 5)$ $(3 , 5 )$ , $(4, 2)$ $(4 ,2 )$ , and $(8, 4)$ $(8 ,4 )$ . What is the radius of the triangle's inscribed circle?

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Answer 1 · 2018-02-08T11:45:54

Radius of triangle's inscribed circle is $1.1$ $1.1$ unit.

Explanation:

The three corners are $A (3, 5) B (4, 2) and C (8, 4)$ $A (3,5) B (4,2) and C (8,4)$

Distance between two points $(x_{1}, y_{1}) and (x_{2}, y_{2})$ $(x_1,y_1) and (x_2,y_2)$ is

$D = \sqrt{{(x_{1} - x_{2})}_{1}^{2} + {(y_{1} - y_{2})}_{1}^{2}}$ $D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2$

Side $A B = \sqrt{{(3 - 4)}^{2} + {(5 - 2)}^{2}} \approx 3.16$ $AB= sqrt ((3-4)^2+(5-2)^2) ~~ 3.16$ unit

Side $B C = \sqrt{{(4 - 8)}^{2} + {(2 - 4)}^{2}} \approx 4.47$ $BC= sqrt ((4-8)^2+(2-4)^2) ~~4.47$ unit

Side $C A = \sqrt{{(8 - 3)}^{2} + {(4 - 5)}^{2}} \approx 5.1$ $CA= sqrt ((8-3)^2+(4-5)^2) ~~ 5.1$ unit

The semi perimeter of triangle is $s = \frac{A B + B C + C A}{2}$ $s=(AB+BC+CA)/2$ or

$s = \frac{3.16 + 4.47 + 5.1}{2} \approx \frac{12.73}{2} \approx 6.37$ $s= (3.16+4.47+5.1)/2~~ 12.73/2~~ 6.37$ unit.

Area of Triangle is $A_{t} = ∣ ∣ ∣ \frac{1}{2} (x 1 (y 2 - y 3) + x 2 (y 3 - y 1) + x 3 (y 1 - y 2)) ∣ ∣ ∣$ $A_t = |1/2(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))|$

$A_{t} = ∣ ∣ ∣ \frac{1}{2} (3 (2 - 4) + 4 (4 - 5) + 8 (5 - 2)) ∣ ∣ ∣$ $A_t = |1/2(3(2−4)+4(4−5)+8(5−2))|$ or

$A_{t} = ∣ ∣ ∣ \frac{1}{2} (- 6 - 4 + 24) ∣ ∣ ∣ = | 7.0 | = 7.0$ $A_t = |1/2(-6-4+24)| = | 7.0| =7.0$ sq.unit.

Incircle radius is $r_{i} = \frac{A_{t}}{s} = \frac{7.0}{6.37} \approx 1.1$ $r_i= A_t/s = 7.0/6.37 ~~1.1$ unit

Radius of triangle's inscribed circle is $1.1$ $1.1$ unit[Ans]

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A triangle has corners at $(3, 5)$ $(3 , 5 )$ , $(4, 2)$ $(4 ,2 )$ , and $(8, 4)$ $(8 ,4 )$ . 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 ), (4,2)(4 ,2 ), and (8,4)(8 ,4 ). What is the radius of the triangle's inscribed circle?

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A triangle has corners at $(3, 5)$ $(3 , 5 )$ , $(4, 2)$ $(4 ,2 )$ , and $(8, 4)$ $(8 ,4 )$ . What is the radius of the triangle's inscribed circle?