Question

in circle o, $overline{ac}$ and $overline{bd}$ are diameters. what is $moverline{bc}$? (3x - 70)° (x + 10)° 50° 80° 100° 130°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, \(3x - 70=x + 10\).

Step2: Solve the equation for \(x\)

Subtract \(x\) from both sides: \(3x-x-70=x - x+10\), which simplifies to \(2x-70 = 10\). Then add 70 to both sides: \(2x-70 + 70=10 + 70\), getting \(2x=80\). Divide both sides by 2: \(x = 40\).

Step3: Find the measure of \(\angle BOC\)

\(\angle BOC\) and \((x + 10)^{\circ}\) are supplementary (linear - pair of angles). Substitute \(x = 40\) into \((x + 10)^{\circ}\), we get \((40+10)^{\circ}=50^{\circ}\). Then \(m\angle BOC=180-(x + 10)\). Since \(x = 40\), \(m\angle BOC=180 - 50=130^{\circ}\). The measure of an arc is equal to the measure of its central angle. So \(m\overset{\frown}{BC}=130^{\circ}\).

Answer:

\(130^{\circ}\)