Question

calculator what is the measure of (overparen{ac})? enter your answer in the box. ( square^{circ} )

Explanation:

Step1: Use the arc - central angle relationship

The measure of an arc is equal to the measure of its central angle. Here, the measure of arc \(AC\) is \(3x + 9\), and the central angle \(\angle ABC=3x - 1.5\). By the inscribed - angle theorem (the measure of an inscribed angle is half the measure of the central angle subtended by the same arc, but here if we assume \(AB\) and \(BC\) are related to the circle properties, and using the fact that for the arc \(AC\), if we consider the relationship between the angle subtended by the arc and the given angles. Wait, actually, the measure of an arc is equal to the measure of its central angle. If we assume that the angle formula: The measure of an arc \(AC\) (let \(m\widehat{AC}=3x + 9\)) and the central angle (if we consider the circle's angle properties, we know that \(m\widehat{AC}=2\times\angle ABC\) (inscribed - angle theorem: an inscribed angle \(\angle ABC\) subtended by an arc \(AC\)). So \(3x + 9=2(3x - 1.5)\)

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(3x+9 = 6x-3\)
Subtract \(3x\) from both sides: \(9=6x - 3-3x\), so \(9 = 3x-3\)
Add \(3\) to both sides: \(9 + 3=3x\), i.e., \(12 = 3x\)
Divide both sides by \(3\): \(x = 4\)

Step3: Find the measure of arc \(AC\)

Substitute \(x = 4\) into \(m\widehat{AC}=3x + 9\)
\(m\widehat{AC}=3\times4+9\)
\(m\widehat{AC}=12 + 9=21\)

Answer:

\(21\)