To solve for the value of [tex]\( x \)[/tex] in the given equations for the central angle [tex]\( AKB \)[/tex] and the angle at the circumference [tex]\( ACB \)[/tex], we will use the inscribed angle theorem. According to this theorem, the central angle subtended by an arc is twice the angle subtended by the same arc at the circumference.
Given:
- Central angle [tex]\( AKB = 4x + 8^\circ \)[/tex]
- Angle at the circumference [tex]\( ACB = 3x - 16^\circ \)[/tex]
According to the inscribed angle theorem:
[tex]\[ \text{Central angle} = 2 \times \text{Angle at the circumference} \][/tex]
Therefore,
[tex]\[ 4x + 8 = 2 \times (3x - 16) \][/tex]
Next, we simplify the equation step-by-step:
1. Expand the right side:
[tex]\[ 4x + 8 = 6x - 32 \][/tex]
2. Move the terms involving [tex]\( x \)[/tex] to one side and the constant terms to the other side:
[tex]\[ 4x - 6x = -32 - 8 \][/tex]
3. Simplify the equation:
[tex]\[ -2x = -40 \][/tex]
4. Divide both sides by [tex]\(-2\)[/tex] to isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-40}{-2} \][/tex]
5. Simplify the division:
[tex]\[ x = 20 \][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( 20 \)[/tex].
