A triangle has corners at $(2, 6)$ $(2 , 6 )$ , $(4, 7)$ $(4 ,7 )$ , and $(1, 5)$ $(1 ,5 )$ . What is the radius of the triangle's inscribed circle?

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Answer 1 · 2017-05-10T08:44:20

Simple. Find the centroid of the triangle and Measure the perpendicular distance from this point to any of the sides.

The centroid of the triangle can be found by the centroid formula:

$x, y$ $x, y$ = $\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}$ $(x_1+x_2+x_3)/3 , (y_1+y_2+y_3)/3$

On plugging values, we get:

$x = \frac{7}{3}$ $x=7/3$

$y = 6$ $y=6$

equation of any one line can be found using the formula:

$y$ $y$ - $y_{1}$ $y_1$ = m( $x$ $x$ - $x_{1}$ $x_1$ )

Where $m$ $m$ =( $y_{2}$ $y_2$ - $y_{1}$ $y_1$ )/( $x_{2}$ $x_2$ - $x_{1}$ $x_1$ )

On plugging in values, we get:

$y - 6 = \frac{x - 2}{2}$ $y - 6 = (x - 2)/2$

$2 y - 12 = x - 2$ $2y - 12 = x - 2$

$x - 2 y - 10 = 0$ $x - 2y - 10 = 0$

The formula for perpendicular distance from a point to a line is:

$d = \frac{| A x + B y + C |}{\sqrt{A^{2} + B^{2}}}$ $d=|Ax + By + C|/sqrt(A^2 + B^2$

On plugging values, we get:

$d = \frac{∣ ∣ \frac{7}{3} - 10 - 10 ∣ ∣}{\sqrt{5}}$ $d = |7/3 - 10 - 10|/sqrt5$

$d = 7.9 c m$ $d = 7.9 cm$

The radius of the circle is $7.9 c m$ $7.9 cm$

A triangle has corners at (2,6)(2 , 6 ), (4,7)(4 ,7 ), and (1,5)(1 ,5 ). What is the radius of the triangle's inscribed circle?