The problem is asking to find the value of $x$ given a central angle and an inscribed angle in a circle. Here is a step-by-step approach to solve it.

Problem Breakdown:

• The circle is centered at point $O$.
• There is an inscribed angle $\angle ABC = 70^\circ$.
• There is a central angle $\angle BOC = 5x - 10$°.
• Since $\angle ABC$ is an inscribed angle, it subtends the same arc as the central angle $\angle BOC$.

Key Concept:

In a circle, the central angle is twice the inscribed angle that subtends the same arc. Therefore: $\angle BOC = 2 \times \angle ABC$

Step-by-Step Solution:

1. Using the relationship between the central and inscribed angles: $5x - 10 = 2 \times 70$
2. Calculate $2 \times 70$: $5x - 10 = 140$
3. Add 10 to both sides: $5x = 150$
4. Divide by 5: $x = 30$

Answer:

The value of $x$ is $30$.

Would you like further details or have any other questions?

Related Questions to Explore:

1. What is the relationship between central and inscribed angles in a circle?
2. How would the solution change if the inscribed angle were different?
3. What if there were multiple inscribed angles subtending the same arc?
4. How can we apply this concept to find unknown angles in cyclic quadrilaterals?
5. What other properties of circles involve inscribed and central angles?

Tip:

Remember that in any circle, the central angle is always double the inscribed angle subtending the same arc.