Question

15 multiple choice 2 points in the figure, \\( \overrightarrow{ba} \\) and \\( \overrightarrow{bc} \\) are tangents of circle \\( o \\). if \\( m \widehat{ac}=140 \\), what is the \\( m \angle b \\)? 140 40 120 90

Explanation:

Step1: Find the measure of the central angle related to \(\angle B\)

The sum of the measures of an arc and its corresponding central angle is \(360^{\circ}\). Given \(m\widehat{AC} = 140^{\circ}\), the central angle \(\angle AOC\) has the same measure as the arc \(AC\), so \(\angle AOC=140^{\circ}\).

Step2: Use the property of tangents and the quadrilateral \(OABC\)

Since \(BA\) and \(BC\) are tangents to the circle \(O\), \(\angle OAB = 90^{\circ}\) and \(\angle OCB=90^{\circ}\). In quadrilateral \(OABC\), the sum of interior angles is \((4 - 2)\times180^{\circ}=360^{\circ}\). Let \(\angle B=x\). Then \(x+\angle OAB+\angle OCB+\angle AOC = 360^{\circ}\). Substituting the known values: \(x + 90^{\circ}+90^{\circ}+140^{\circ}=360^{\circ}\).

Step3: Solve for \(x\) (which is \(\angle B\))

Simplify the equation \(x+320^{\circ}=360^{\circ}\). Subtract \(320^{\circ}\) from both sides: \(x=\angle B=360^{\circ}- 320^{\circ}=40^{\circ}\).

Answer:

\(40\) (the second option)