From the provided image, it appears to be a mathematical problem involving a triangle (ABC) and its rotation about a point $O$ by 70 degrees counterclockwise, resulting in a new triangle $\triangle PQR$. Let me explain and provide some steps for solving it:

Problem Breakdown:

1. Part 1: Sketch $\triangle PQR$:

   • Rotate $\triangle ABC$ by 70 degrees counterclockwise about point $O$. Each vertex (A, B, C) is rotated to its new corresponding position (P, Q, R) using geometric rotation rules.

2. Part 2: Identify the line segment equal in length to $OA$:

   • Determine which line in $\triangle PQR$ corresponds in length to $OA$, as distances from $O$ to the vertices remain constant during rotation.

3. Part 3: Find angles equal to $\angle AOP$:

   • Identify two other angles in the problem (from either $\triangle ABC$ or $\triangle PQR$) that are congruent to $\angle AOP$, as rotation preserves angles between corresponding points.

Approach to Solve:

1. Part 1: To draw $\triangle PQR$:

   • Rotate $A, B, C$ counterclockwise by 70 degrees. You can use the rotation formula: x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$ where $$(x, y)$$ are the original coordinates, and $$\theta = 70^\circ$$.

2. Part 2: Line segments from $O$ remain unchanged in length. For example, if $OA = OP$, then $OP$ is equal in length to $OA$.

3. Part 3: Use geometric properties and rotational symmetry to identify congruent angles.

Would you like detailed calculations, diagrams, or step-by-step explanations for any part? Let me know! Here are some related questions to deepen your understanding:

Related Questions:

1. How do you derive the rotation matrix formula in coordinate geometry?
2. What are the properties of triangles after rotation in a plane?
3. How do you calculate the length of a line segment using the distance formula?
4. What is the relationship between the angles in congruent triangles?
5. Can you describe the transformation of geometric shapes under rotation?

Tip:

Always verify the length and angle relationships using symmetry properties when dealing with rotational transformations.