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The image contains several geometry problems involving quadrilaterals. I'll go through each one and provide solutions or guidance where needed.

### 1. **Problem 1**: A rhombus with diagonals AC = 16 cm and BD = 12 cm.
   
   The task seems to be related to properties of rhombuses. Diagonals in a rhombus bisect each other at right angles and form four right-angled triangles.

   To calculate the side of the rhombus:
   \[
   \text{Side of rhombus} = \sqrt{\left(\frac{AC}{2}\right)^2 + \left(\frac{BD}{2}\right)^2}
   \]
   \[
   = \sqrt{\left(\frac{16}{2}\right)^2 + \left(\frac{12}{2}\right)^2}
   = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \, \text{cm}
   \]
   So, the side length is 10 cm.

---

### 2. **Problem 2**: A rectangle PQRS with PT = TS = 5 cm and PS = 13 cm.

   This suggests that triangle PTS is a right-angled triangle. By applying the Pythagorean theorem, you can find the length of the diagonal PS.
   \[
   PS^2 = PT^2 + TS^2
   \]
   Solving for the sides of the rectangle would involve recognizing that PT = TS is a known condition.

---

### 3. **Problem 3**: A square with each angle equal to 90°. The side length is 8 cm.

   This seems to be a standard square problem. The area of the square can be calculated as:
   \[
   \text{Area} = 8 \times 8 = 64 \, \text{cm}^2
   \]
   Additionally, the diagonal can be calculated using the formula for the diagonal of a square:
   \[
   \text{Diagonal} = \text{side} \times \sqrt{2} = 8 \times \sqrt{2} \approx 11.31 \, \text{cm}
   \]

---

### 4. **Problem 4**: A rectangle with AB = 9 cm and BC = 4 cm. Find the value of x.

   Likely, this involves the Pythagorean theorem if you are looking for a diagonal or another unknown side:
   \[
   \text{Diagonal}^2 = AB^2 + BC^2
   \]
   Substituting the known values:
   \[
   \text{Diagonal} = \sqrt{9^2 + 4^2} = \sqrt{81 + 16} = \sqrt{97} \approx 9.85 \, \text{cm}
   \]

---

### 5. **Problem 5**: A rhombus with CF = 4 cm and diagonals given. The task is to calculate the length of the other diagonal.

   As with the first problem, you can use the Pythagorean theorem, as diagonals bisect each other perpendicularly in a rhombus.

---

Would you like more detailed explanations on any of these problems or further clarification? Here are five related questions that might help expand your understanding:

1. How do you calculate the area of a rhombus using its diagonals?
2. What is the formula for the diagonal of a square?
3. How can the Pythagorean theorem be applied to different quadrilaterals?
4. What are the properties of diagonals in a rhombus?
5. How do you derive the side length of a rectangle given its diagonal?

**Tip**: Always keep in mind the unique properties of quadrilaterals, such as how diagonals behave in shapes like squares, rectangles, and rhombuses. They often provide the key to solving problems involving these shapes.

The image contains several geometry problems involving quadrilaterals. These include calculating sides, diagonals, and angles in rhombuses, squares, and rectangles.

This page provides solutions for geometry problems involving rhombuses, squares, and rectangles. Key concepts include the Pythagorean theorem, properties of quadrilaterals, and calculating sides and diagonals. Suitable for students in Grades 8-10.

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