Question

1. find x.

x =

Explanation:

Step1: Use the property of a circle's central angles

The sum of central angles in a circle is \(360^{\circ}\). Here, \(AB\) is a diameter, so the sum of \(\angle ADC\) and \(\angle BDC\) related to the semicircle is \(180^{\circ}\).

Step2: Set up the equation

We know that \(\angle ADC=(10x - 23)^{\circ}\) and \(\angle BDC = 106^{\circ}\). Since \(\angle ADC+\angle BDC = 180^{\circ}\), we have the equation \(10x-23 + 106=180\).

Step3: Simplify the equation

First, combine like - terms: \(10x+( - 23 + 106)=180\), which gives \(10x + 83=180\).
Then, subtract 83 from both sides: \(10x=180 - 83\), so \(10x=97\).

Step4: Solve for \(x\)

Divide both sides by 10: \(x=\frac{97}{10}=9.7\).

Answer:

\(x = 9.7\)