Math Problem Statement
Solve
Solution
The problem shows a circle with center , and asks to find the value of . The given conditions are:
I'll walk through the steps to solve this geometrically.
Step-by-Step Solution:
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Identifying the Angles:
- (this is an angle at the circumference).
- (another angle at the circumference).
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Understanding the Circle Theorems:
- The angles subtended by the same arc at the circumference of a circle are equal.
- The sum of the opposite angles in a cyclic quadrilateral is .
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Finding :
- Notice that is subtended by the same arc as .
- Therefore, by the circle theorem, .
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Finding :
- and are opposite angles in a cyclic quadrilateral, meaning their sum should be .
- Therefore, .
- Substituting , we get:
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Calculating :
- Now, and .
- So, .
Final Answer:
The value of is .
Do you have any questions about this solution? Would you like further clarification?
Here are some related questions that might help you explore this topic further:
- What are the key properties of cyclic quadrilaterals?
- How do you use circle theorems to solve geometric problems?
- What is the relationship between the angles subtended by the same arc in a circle?
- How would you solve this problem if the diagram involved a different set of angles?
- What is the significance of opposite angles in cyclic quadrilaterals?
Tip: When working with circle geometry, always check for opportunities to apply circle theorems, as they often simplify the problem significantly.
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Math Problem Analysis
Mathematical Concepts
Geometry
Circle Geometry
Circle Theorems
Formulas
-
Theorems
Circle theorems
Suitable Grade Level
Grades 7-9