Let's solve the questions step by step from the image you uploaded:

1. Circle Geometry - Angles C and D

Given: $\angle AOB = 48^\circ$

• Angle C:
  Since $\angle AOB$ is a central angle and subtended by arc $AB$, the inscribed angle $\angle AC = \frac{1}{2} \angle AOB = \frac{1}{2} \times 48^\circ = 24^\circ$.

• Angle D:
  The angle $\angle D$ subtended by the arc from $B$ to $A$ on the circumference is the same as $\angle C$, so $\angle D = 24^\circ$.

Therefore:
$\angle C = 24^\circ$,
$\angle D = 24^\circ$.

2. Find the value of $x$ in the second circle diagram

Given: $\angle ACB = (x + 50)^\circ$, and we know that the total sum of the angles in a circle is $360^\circ$.

• The exterior angle $(380 - 4x)^\circ$ is related to the interior opposite angles by the formula: $\text{Exterior Angle} = 180^\circ - \text{Interior Opposite Angle}$

  Let's assume this relationship and solve for $x$. I'll compute that next.

3. Triangle Geometry - Equilateral Triangle STU

Given: $\triangle STU$ is equilateral, so $\angle STU = 60^\circ$ for each interior angle of the triangle.

Some specific values are already provided for different angles. Let's fill in the remaining ones:

• Angle VPU: Given as $98^\circ$
• Angle TPU: Given as $82^\circ$
• Angle SPV: Given as $82^\circ$
• Angle TVU: Since $\triangle STU$ is equilateral, each angle at the center involving $T, V, U$ will also be $60^\circ$.

Therefore:

• $\angle TVU = 60^\circ$
• $\angle SVT = 60^\circ$
• $\angle SUV = 60^\circ$

4. Finding the radius of the circle (center at (13,0))

The radius can be found using the distance formula between the center of the circle and any point on the circumference. The center is given as $(13, 0)$, and a point on the circle is $(0, 9)$.

• The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the values: $r = \sqrt{(13 - 0)^2 + (0 - 9)^2} = \sqrt{13^2 + 9^2} = \sqrt{169 + 81} = \sqrt{250}$ Therefore, the radius is $\sqrt{250} \approx 15.81$ units.

Let me know if you would like more details on any step or if you'd like me to assist with the remaining calculations. Here are 5 relative questions for further learning:

1. How do you calculate angles subtended by chords in a circle?
2. What is the relation between central angles and inscribed angles in a circle?
3. How is the distance formula used to calculate the radius in coordinate geometry?
4. Can you describe how an equilateral triangle's internal and external angles relate?
5. How do you find the measure of an angle formed by intersecting chords?

Tip: When dealing with inscribed angles, remember that the inscribed angle is always half the measure of the central angle subtended by the same arc!