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The image contains a geometric problem with multiple parts related to a cyclic quadrilateral \( QRST \) inscribed in a circle. Below is a summary of the questions and the steps needed to solve them.

### Given Information:
- \( AB \) is a diameter of the circle.
- \( QP, RS, \) and \( QT \) are tangents to the circle at points \( A, C, \) and \( D \) respectively.
- \( AP = AQ \).

### Questions and Solutions:

**Q21**: The measure of \( \angle ACB = 57^\circ \). Calculate and write the measure of \( \angle QRC \).
- **Solution**: Since \( AB \) is the diameter and \( \angle ACB \) subtends the diameter, \( \angle ACB = 90^\circ \). Given \( \angle ACB = 57^\circ \), \( \angle QRC = 90^\circ - 57^\circ = 33^\circ \).

**Q22**: Is triangle \( \angle SRQ \) a right-angled triangle? Justify your answer.
- **Solution**: \( \angle SRQ \) is not a right-angled triangle because it was determined \( \angle QRC = 33^\circ \).

**Q23**: The sum of the opposite sides of a quadrilateral inscribed in a circle equals the sum of two sides. Justify your answer.
- **Solution**: This is true for any cyclic quadrilateral due to Ptolemy's Theorem. Therefore, \( QS + RT = QR + ST \).

**Q24**: The length of \( SR = 5 \) cm, \( RQ = 6 \) cm, \( QT = 7 \) cm, and \( ST = 8 \) cm. What is the perimeter of quadrilateral \( QRST \)?
- **Solution**: The perimeter is calculated as \( SR + RQ + QT + ST = 5 + 6 + 7 + 8 = 26 \) cm.

### Any questions, or would you like further details?

### Related Questions:
1. How does Ptolemy's Theorem apply to cyclic quadrilaterals?
2. What are the conditions for a quadrilateral to be inscribed in a circle?
3. Can the diagonals of a cyclic quadrilateral be used to find the area?
4. How does the Inscribed Angle Theorem relate to the problem?
5. What are the properties of a tangent to a circle?

### Tip:
When dealing with cyclic quadrilaterals, always consider properties like opposite angles summing to 180° and the application of Ptolemy's Theorem for solving side lengths.

The image contains a geometric problem with multiple parts related to a cyclic quadrilateral QRST inscribed in a circle. The given information includes that AB is a diameter of the circle, and QP, RS, and QT are tangents to the circle at points A, C, and D, respectively. AP = AQ. The problems involve calculating angles, identifying right-angled triangles, justifying the properties of cyclic quadrilaterals, and calculating the perimeter of the quadrilateral.

Solve a detailed geometric problem involving a cyclic quadrilateral QRST inscribed in a circle. Learn how to apply the Inscribed Angle Theorem and Ptolemy's Theorem, and calculate the perimeter of the quadrilateral. Suitable for students in Grades 9-11.

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