Question

in circle b, ( moverarc{dc} = 109^circ ). solve for ( x ) if ( mangle dbc = (9x + 24)^circ ). if necessary, round your answer to the nearest tenth.

Explanation:

Step1: Recall central angle theorem

The measure of a central angle is equal to the measure of its intercepted arc. So \( m\angle DBC = m\widehat{DC} \).

Step2: Set up the equation

We know \( m\widehat{DC}=109^\circ \) and \( m\angle DBC=(9x + 24)^\circ \), so we set up the equation:
\( 9x + 24 = 109 \)

Step3: Solve for x

Subtract 24 from both sides:
\( 9x = 109 - 24 \)
\( 9x = 85 \)
Divide both sides by 9:
\( x=\frac{85}{9}\approx9.4 \)

Answer:

\( x\approx9.4 \)