Let's analyze and solve the problems one by one from the image you provided. These are triangle problems involving proportions or the Pythagorean theorem. Here's the solution for each question:

Question 9

The given triangle has two segments forming a right triangle, divided into segments 15, 14, and 21. To find $x$, we apply the property of proportions in a right triangle using similar triangles.

Using the similarity property: $\frac{15}{x} = \frac{x}{14}$

Cross-multiplying: $15 \cdot 14 = x^2$

Simplify: $x^2 = 210 \quad \implies \quad x = \sqrt{210}$

Approximating: $x \approx 14.49$

Question 10

The triangle involves a height and the use of the geometric mean theorem. To solve for $y$, use: $\frac{4}{y} = \frac{y}{6}$

Cross-multiply: $4 \cdot 6 = y^2$

Simplify: $y^2 = 24 \quad \implies \quad y = \sqrt{24}$

Approximating: $y \approx 4.9$

Question 11

The triangle is solved using proportions. The variables 3, $z$, and 4.5 relate to similar triangles. Use: $\frac{z}{4.5} = \frac{3}{1.5}$

Cross-multiply: $z \cdot 1.5 = 3 \cdot 4.5$

Simplify: $1.5z = 13.5 \quad \implies \quad z = \frac{13.5}{1.5}$

Simplify further: $z = 9$

Final Answers:

1. $x \approx 14.49$
2. $y \approx 4.9$
3. $z = 9$

Would you like additional explanations or step-by-step breakdowns for any of these?

Here are 5 related questions for further learning:

1. How can you verify similarity in triangles before solving proportional problems?
2. What are other ways to use the geometric mean theorem in triangle geometry?
3. How can you derive the Pythagorean theorem from basic geometric principles?
4. What is the importance of proportions in solving right triangle problems?
5. How does solving for variables differ in obtuse versus right triangles?

Tip: Always double-check your final answers by substituting back into the original proportions!