### Textual Information #### Question Stem: Point \( P \) is the center of the circle in the figure above. What is the value of \( x \)? #### Other Relevant Text: - The points \( A \), \( B \), and \( C \) lie on the circumference of the circle. - The angles at points \( A \), \( B \), and \( C \) are each labeled as \( 20^\circ \). - \( P \) is the center of the circle, and line segments \( AP \), \( BP \), and \( CP \) are radii of the circle. #### Mathematical Formulas/Chemical Equations: No specific formulas or chemical equations are present in the image. #### Options: No options are provided in the image. ### Chart/Diagram Description #### Type: The chart is a geometric figure involving a circle with specific points, lines, and angles. #### Main Elements: ##### Points: - **\( P \):** The center of the circle. - **\( A \), \( B \), and \( C \):** Points lying on the circumference of the circle. ##### Lines: - **\( AP \), \( BP \), and \( CP \):** These are radii of the circle, connecting the center \( P \) to the points \( A \), \( B \), and \( C \) on the circumference. - **\( AB \), \( BC \), and \( CA \):** These are chords of the circle, connecting the points on the circumference. ##### Shapes: - **Circle:** The main shape in the diagram, with \( P \) as its center. - **Triangles:** Multiple triangles are formed, including \( \triangle APB \), \( \triangle BPC \), and \( \triangle CPA \). ##### Angles: - **\( \angle PAB = 20^\circ \):** The angle at point \( A \) in \( \triangle APB \). - **\( \angle PBC = 20^\circ \):** The angle at point \( B \) in \( \triangle BPC \). - **\( \angle PCA = 20^\circ \):** The angle at point \( C \) in \( \triangle CPA \). ##### Labels and Annotations: - **\( x^\circ \):** The angle at point \( P \) in \( \triangle APB \), which is the value to be determined. - **Equal Length Marks:** Indicate that \( AP \), \( BP \), and \( CP \) are of equal length (radii of the circle). ##### Relative Position and Direction: - \( A \), \( B \), and \( C \) are positioned on the circumference of the circle. - \( P \) is centrally located, with radii extending to \( A \), \( B \), and \( C \). - The angles at \( A \), \( B \), and \( C \) are each \( 20^\circ \), and they are subtended by the respective radii. ### Output Format **Question Stem:** Point \( P \) is the center of the circle in the figure above. What is the value of \( x \)? **Other Relevant Text:** The points \( A \), \( B \), and \( C \) lie on the circumference of the circle. The angles at points \( A \), \( B \), and \( C \) are each labeled as \( 20^\circ \). \( P \) is the center of the circle, and line segments \( AP \), \( BP \), and \( CP \) are radii of the circle. **Chart/Diagram Description:** The chart is a geometric figure involving a circle with specific points, lines, and angles. - **Points:** \( P \) (center), \( A \), \( B \), and \( C \) (on the circumference). - **Lines:** Radii \( AP \), \( BP \), and \( CP \); chords \( AB \), \( BC \), and \( CA \). - **Shapes:** Circle and multiple triangles (\( \triangle APB \), \( \triangle BPC \), \( \triangle CPA \)). - **Angles:** \( \angle PAB = 20^\circ \), \( \angle PBC = 20^\circ \), \( \angle PCA = 20^\circ \). - **Labels:** \( x^\circ \) (angle at \( P \) in \( \triangle APB \)). **Mathematical Analysis for \( x \):** Since \( \angle PAB = 20^\circ \) and \( \angle PBA = 20^\circ \) (isosceles triangle properties), the angle \( \angle APB = x \) can be found using the triangle angle sum property: \[ x + 20^\circ + 20^\circ = 180^\circ \] \[ x + 40^\circ = 180^\circ \] \[ x = 140^\circ \] Thus, the value of \( x \) is \(\boxed{140}\).