Solve for x in Circle Inscribed Angle Problem

Answer

Based on the image provided, the problem asks you to solve for $x$ given an inscribed angle $\angle EBC$ and its intercepted arc $\stackrel{\frown}{EC}$. By applying the Inscribed Angle Theorem, the value of $x$ is found to be $20.8$.

Explanation

The image shows a circle with center $D$ and points $E, B, C$ on the circumference. The angle $\angle EBC$ is an inscribed angle because its vertex $B$ lies on the circle and its sides are chords. The arc intercepted by this angle is $\stackrel{\frown}{EC}$.

Known Quantities:

• Measure of intercepted arc $\stackrel{\frown}{EC} = 120^{\circ}$
• Measure of inscribed angle $\angle EBC = (5x - 44)^{\circ}$

1. Identify the Relationship The Inscribed Angle Theorem states that the measure of an inscribed angle is exactly half the measure of its intercepted arc. $m\angle EBC = \frac{1}{2} \cdot m\stackrel{\frown}{EC}$ This formula expresses that an angle whose vertex is on the circle is half the size of the rotational distance between its endpoints on the arc. ⚠️ This step is required on exams to justify your equation.

2. Substitute Known Values
   Substitute the expressions provided in the problem into the geometric theorem. $5x - 44 = \frac{1}{2} \cdot 120$ We replace the variable names with the specific algebraic and numerical values shown in the diagram.

3. Simplify the Equation Calculate the right side of the equation to simplify the constant term. $5x - 44 = 60$ Half of $120$ is $60$, which represents the constant value of the inscribed angle in degrees.

4. Isolate the Variable Term Add $44$ to both sides of the equation to move the constant term to the right. $5x = 104$ By adding $44$ to $60$, we determine that $5$ times our unknown value equals $104$.

5. Solve for $x$ Divide both sides by $5$ to find the numerical value of $x$. $x = \frac{104}{5}$ Division is used here to undo the multiplication of the coefficient $5$.

6. Decimal Conversion Since the problem asks for the nearest tenth if necessary, convert the fraction to a decimal. $x = 20.8$ The value $20.8$ is the precise decimal quotient of $104$ divided by $5$.

Final Answer

The value of $x$ is: $\boxed{20.8}$

Common Mistakes

• Confusing Inscribed and Central Angles: Students often set the angle equal to the arc (e.g., $5x - 44 = 120$). This is only true if the vertex is at the center ($D$). For vertices on the circle ($B$), you must divide the arc by $2$.
• Rounding Errors: Always check the instructions for rounding. In this case, $20.8$ is an exact decimal, so no further rounding is required to reach the "nearest tenth."