Question

point o is the incenter of \\(\delta abc\\).
what is \\(m\angle qbo\\)?
\\((3x - 15)^\circ\\)
\\((2x + 6)^\circ\\)
\\((4x - 12)^\circ\\)

• \\(5^\circ\\)
• \\(9^\circ\\)
• \\(12^\circ\\)
• \\(24^\circ\\)

Explanation:

Identify the properties of the incenter

The incenter of a triangle is the intersection point of the triangle's three angle bisectors. Since point \(O\) is the incenter of \(\Delta ABC\), the segment \(AO\) bisects \(\angle BAC\), and the segment \(BO\) bisects \(\angle ABC\).

Set up the equation for x

Since \(AO\) is the angle bisector of \(\angle BAC\), the two angles \(\angle QAO\) and \(\angle SAO\) are equal:

$$m\angle QAO = m\angle SAO$$

Substitute the given expressions:

$$2x + 6 = 4x - 12$$

Solve for x

Rearrange the equation to solve for \(x\):

$$6 + 12 = 4x - 2x$$

$$18 = 2x$$

$$x = 9$$

Calculate the measure of angle QBO

Since \(BO\) is the angle bisector of \(\angle ABC\), the angle \(\angle QBO\) is represented by the expression \((3x - 15)^\circ\). Substitute \(x = 9\) into the expression:

$$m\angle QBO = 3(9) - 15$$

$$m\angle QBO = 27 - 15$$

$$m\angle QBO = 12^\circ$$

Answer:

• 5°
• 9°
• 12° (Correct answer)
• 24°