Math Problem Statement
Solution
Thank you for the clearer image. Let's break down the problem step-by-step based on the diagram.
Diagram Analysis:
- is the topmost point, and , are endpoints of the diameter.
- is the center of the semicircle.
- and are points on the semicircle.
- Given:
We need to determine .
Step-by-Step Solution:
-
Understanding :
- is formed at the center of the circle by the radii and . This implies that the arc subtended by measures .
-
Using Properties of Inscribed Angles:
- Inscribed angles subtending the same arc are half the measure of the central angle. Therefore, (which subtends the same arc) is .
-
Analyzing :
- In a semicircle, the angle subtended by a diameter is . Hence, in triangle is .
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Calculating :
- subtends arc , which is opposite . The key here is to understand the relationship between the angles.
-
Adding the Angles:
- We are asked to find .
After analyzing these relationships:
- and sum up to .
Final Answer:
The correct choice is:
(C) 150°
Would you like a more detailed explanation or have any other questions? Here are 5 questions related to this topic that you might find interesting:
- How do you determine the arc length given an inscribed angle?
- What is the relationship between central angles and inscribed angles in a circle?
- How can you determine the radius of a semicircle given a chord and an inscribed angle?
- What is the significance of the angle subtended by a diameter in different polygons?
- How does the angle sum property apply in cyclic quadrilaterals?
Tip: Remember that in any circle, the angle subtended by a diameter is always .
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Math Problem Analysis
Mathematical Concepts
Geometry
Circle Geometry
Inscribed Angles
Semicircles
Formulas
-
Theorems
Angle sum property of triangles
Properties of inscribed angles
Suitable Grade Level
High School
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