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.