Question

I is the incenter of the triangle,AI=3x+7, BI=5x-11, & CI=52-2xAI=[?]

Answer 1

Answer:

Given that,

I is the incenter of thr triangle.

A, B and C are points on the each side of the triangle (foot of the perpendicular bisector of each side of the triangle).

AI= 3x+7

BI= 5x-11

CI= 52-2x

we know that,

The incircle is a circle inscribed in the triangle (polygon), and the centre of the circle is the point of intersection of the angular bisectors of the triangle (polygon).

we can draw an incircle with I as center that passes through A, B and C.

we get that, AI, BI and CI are radius and all are equal, that is,

$AI=BI=CI$

we have that,

$\begin{gathered} AI=BI \\ 3x+7=5x-11 \end{gathered}$

$5x-3x=11+7$

$2x=18$

$x=9$

Substitute x=9 in AI, we get

$AI=3x+7=34$

Radius of the incircle is 34 units.

Verification:

Substitute x=9 in CI, we get,

$\begin{gathered} CI=52-2(9) \\ =52-18 \\ CI=34 \end{gathered}$

AI=CI.

Hence it is verified.

we get that,

$AI=34$

Answer is:

$AI=34$