The measure of an inscribed angle is $x$, and the measure of its intercepted arc is $y$. Which equation describes the relationship between $x$ and $y$?
A. $y=180-2 x$
B. $x=y$
C. $y=2 x$
D. $x=2 y$
Answer (2)
Recalls the inscribed angle theorem.
Expresses the relationship between the inscribed angle x and its intercepted arc y as x = 2 1 y .
Solves for y in terms of x , resulting in y = 2 x .
Concludes that the correct equation is y = 2 x .
Explanation
Understanding the Inscribed Angle Theorem Let's analyze the relationship between an inscribed angle and its intercepted arc. The inscribed angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.
Expressing the Relationship Let x be the measure of the inscribed angle and y be the measure of its intercepted arc. According to the inscribed angle theorem, we have: x = 2 1 y
Solving for y Now, we want to express y in terms of x . To do this, we multiply both sides of the equation by 2: 2 x = 2 × 2 1 y 2 x = y So, we have: y = 2 x
Identifying the Correct Option Comparing this equation with the given options, we find that option C matches our result.
Final Answer Therefore, the equation that describes the relationship between x and y is y = 2 x .
Examples
In architecture, understanding inscribed angles and their intercepted arcs is crucial when designing arched windows or circular structures. For example, if an architect wants an arched window to provide a specific view angle ( x ), they need to calculate the corresponding arc ( y ) of the window's curve to ensure the desired field of vision is achieved. The relationship y = 2 x helps them accurately determine the curvature needed for the window design.
The correct equation that describes the relationship between the inscribed angle x and its intercepted arc y is y = 2 x . The inscribed angle theorem states that the measure of the inscribed angle is half of the intercepted arc. Therefore, the answer is option C, y = 2 x .
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