Question

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

Explanation:

Step1: Use vertical - angle property

Since $\angle AOB$ and $\angle DOC$ are vertical angles, $3x - 70=x + 10$.

Step2: Solve the equation for $x$

$3x-x=10 + 70$, $2x=80$, $x = 40$.

Step3: Find $\angle AOB$

Substitute $x = 40$ into $\angle AOB=3x - 70$, we get $\angle AOB=3\times40-70=120 - 70 = 50^{\circ}$.

Step4: Find $\angle BOC$

Since $\angle AOB+\angle BOC = 180^{\circ}$ (linear - pair of angles), $\angle BOC=180^{\circ}-\angle AOB$. So $\angle BOC=180 - 50=130^{\circ}$. The measure of an arc is equal to the measure of its central angle, so $m\overparen{BC}=130^{\circ}$.

Answer:

$130^{\circ}$