Question

point o is the incenter of \\( \triangle abc \\). what is \\( m\angle qbo \\)? \\( 5 ^ { \circ } \\) \\( 9 ^ { \circ } \\) \\( 12 ^ { \circ } \\) \\( 24 ^ { \circ } \\)

Explanation:

Step1: Use the property of in - center (angle bisector)

Since \(O\) is the in - center of \(\triangle ABC\), \(AO\) is the angle bisector of \(\angle BAC\). So \(2x + 6=4x-12\).
Solve the equation \(2x + 6=4x-12\):
Subtract \(2x\) from both sides: \(6 = 2x-12\).
Add \(12\) to both sides: \(2x=18\), then \(x = 9\).

Step2: Calculate \(\angle QBO\)

Substitute \(x = 9\) into the expression for \(\angle QBO=(3x - 15)^{\circ}\).
\(\angle QBO=(3\times9 - 15)^{\circ}=(27 - 15)^{\circ}=12^{\circ}\).

Answer:

\(12^{\circ}\) (the third option)