The image you've uploaded contains a series of geometric problems involving forming and solving equations. Below, I will provide a detailed breakdown of each problem and how they were solved.

1. Triangle Perimeter

• Expression: $P = 3x + 2x + 4x = 9x$
• Equation: Given $P = 27$ cm, solve $9x = 27$.
• Solution: $x = \frac{27}{9} = 3$ cm

2. Quadrilateral Perimeter

• Expression: $P = (w + 3) + (2w + 3) + (w + 1) + 2w = 6w + 7$
• Equation: Given $P = 31$ cm, solve $6w + 7 = 31$.
• Solution: $6w = 31 - 7 = 24$, so $w = \frac{24}{6} = 4$ cm

3. Isosceles Triangle Perimeter

• Expression: $P = 4z + 2z + 4z + 2 = 8z + 2$
• Equation: Given $P = 28$ cm, solve $8z + 12 = 28$.
• Solution: $8z = 28 - 12 = 16$, so $z = \frac{16}{8} = 2$ cm

4. Triangle Perimeter

• Expression: $P = 5y + 4(y + 3) + 3y = 5y + 4y + 12 + 3y = 12y + 12$
• Equation: Given $P = 30$ cm, solve $12y + 12 = 30$.
• Solution: $12y = 30 - 12 = 18$, so $y = \frac{18}{12} = 1.5$ cm

5. Rectangle Equation

• Equation: Solve $4x + 2 = 2x + 7$.
• Solution: $4x - 2x = 7 - 2$, so $2x = 5$ and $x = 2.5$ cm
• Perimeter: $P = 2 \times (4x + 2) = 2 \times (10 + 2) = 24$ cm

6. Irregular Pentagon Perimeter

• Expression: $P = 4x - 4 + 15 - 2x + 2(2x - 2) + 3x - 6$
• Equation: Simplify and solve for $x$.
• Solution: Perimeter equation was solved as $x = 2.5$ cm

7. Quadrilateral Interior Angles

• Equation: $8x + 160 = 360$ (Sum of interior angles)
• Solution: $8x = 200$, so $x = 25^\circ$

8. Rectangle Area

• Given: Width $= x$, Length $= 3x$, Perimeter $= 72$ cm
• Equation: $2x + 6x = 72$, solve for $x$.
• Solution: $x = 9$ cm, Area $= 9 \times 27 = 243$ cm²

9. Rectangle Areas

• Rectangle A: $A = 6 \times (x + 1)$, $B = 3 \times (3x + 7)$
• Area Relationship: $A_B = 2 \times A_A$, calculate individual areas.

Would you like more detailed explanations or do you have any specific questions? Here are five related questions to explore:

1. How can the method for solving perimeter problems be applied to other shapes like circles?
2. What are the steps to finding the area of composite shapes that combine rectangles and triangles?
3. How would the approach change if the measurements involved variables with exponents?
4. Can you derive the general formula for the sum of interior angles of any polygon?
5. How does solving these equations differ when considering three-dimensional figures?

Tip: When solving for unknowns in geometric problems, carefully define expressions based on the shape’s properties (e.g., perimeter or area) and ensure all sides or angles are correctly accounted for before forming an equation.