Find m∠QPR in Circle: Inscribed Angle Solution

Answer

The measure of $\angle QPR$ is $74^\circ$. This result is found by using the Inscribed Angle Theorem to relate the angle to its intercepted arc and using the fact that a full circle sums to $360^\circ$.

Explanation

Based on the image provided, we observe a circle with three points on its circumference: $P$, $Q$, and $R$.

• Arc $PQ$ is labeled as $8x - 9$.
• Arc $PR$ is labeled as $11x + 3$.
• Inscribed Angle $QPR$ is labeled as $5x + 9$.
• Arc $QR$ is the intercepted arc for the inscribed angle $\angle QPR$.

1. Relate the Inscribed Angle to its Intercepted Arc According to the Inscribed Angle Theorem, the measure of an angle inscribed in a circle is exactly half the measure of its intercepted arc. $m\angle QPR = \frac{1}{2} m\text{arc } QR$ This formula establishes that arc $QR$ is twice the size of the angle $\angle QPR$.

   ⚠️ This step is required on exams to justify the relationship between arcs and angles.

2. Express Arc $QR$ in terms of $x$ Since $m\angle QPR = 5x + 9$, we can multiply by 2 to find the expression for the intercepted arc $QR$. $m\text{arc } QR = 2(5x + 9) = 10x + 18$ This expression represents the portion of the circle's circumference bounded by points $Q$ and $R$.

3. Set up the Sum of the Circle The sum of all arcs in a circle must equal $360^\circ$. We add the expressions for arc $PQ$, arc $PR$, and arc $QR$ together. $(8x - 9) + (11x + 3) + (10x + 18) = 360$ This equation accounts for every degree around the center of the circle using the algebraic labels provided.

4. Solve for $x$ Combine like terms and isolate the variable $x$. $29x + 12 = 360$ $29x = 348$ $x = \frac{348}{29} = 12$ This calculation determines the scale factor $x$ needed to find the actual geometric measurements.

5. Calculate the Measure of $\angle QPR$ Now, substitute the value $x = 12$ back into the original expression for the angle. $m\angle QPR = 5(12) + 9$ $m\angle QPR = 60 + 9$ $m\angle QPR = 69^\circ$ Correction Note: Re-evaluating the arithmetic: $5 \times 12 = 60$, and $60 + 9 = 69$. Let's ensure the substitution is precise.

   Wait, let's re-verify the sum: $8+11+10 = 29$. $-9+3+18 = 12$. $360-12=348$. $348/29 = 12$. Calculated Angle: $5(12)+9 = 69^\circ$.

6. Unit Check The final value is an angle measurement, so the units are degrees. $60 + 9 = 69^\circ$ This value represents the opening of angle $QPR$.

Final Answer

$\boxed{69^\circ}$

Common Mistakes

• Confusing Arc and Angle: Students often set the angle equal to the arc (e.g., $5x + 9 = \text{arc } QR$) instead of doubling the angle to find the arc. Always remember: $\text{Arc} = 2 \times \text{Angle}$.
• Incomplete Sum: Forgetting to include the "missing" arc $QR$ when setting the total equal to $360^\circ$. You must express all three parts of the circle's circumference.