Question

Point A is the incenter of triangle PQR. Find each answer. Justify your answer for A and B.

Answer 1

The incenter is defined as the center point of a triangle formed by all three angle bisectors.

An important characteristic of an incenter is that it is equidistant to each side.

To find angle ARU, we have to notice that RA is a bisector, so we can say

`\begin{gathered} m\angle ARU=m\angle ARK \\ m\angle ARU=40 \end{gathered}`

Therefore, the measure of angle ARU is 40°.

Additionally, side AU is equal to TA because they are equidistant to the incenter, by definition. So,

`AU=TA=20`

Therefore, side AU is 20 units long.

On the other hand, Angle QPA can be found with the expression

`m\angle QPA=m\angle RPA`

Since angle P is being bisected, so using the given expressions we have

`3x+2=4x-9`

We solve for x

`\begin{gathered} 3x-4x=-9-2 \\ -x=-11 \\ x=11 \end{gathered}`

So, angle QPA would be

`m\angle QPA=3x+2=2(11)+2=33+2=35`

Therefore, angle QPA is 35°.

At last, angle KQA would be half of angle Q due to the bisector QA. So, we can define the following

`m\angle Q=180-70-80=30`

And,

`m\angle KQA=(m\angle Q)/(2)=15`

Therefore, angle KQA is 15°.