In the image, you have two similar triangles, PQR and ABC. The sides are labeled as follows:

• For triangle PQR:

  • $PQ = 20$
  • $PR = b$
  • $QR = a$

• For triangle ABC:

  • $AB = 9$
  • $AC = 12$
  • $BC = 15$

Since the triangles are similar, we can use the property of similar triangles that the corresponding sides are proportional. Therefore, we can set up the following proportions:

$\frac{PQ}{AB} = \frac{QR}{BC} = \frac{PR}{AC}$

Now, substitute the known values:

$\frac{20}{9} = \frac{a}{15} = \frac{b}{12}$

Step 1: Solve for $a$

Using the proportion $\frac{20}{9} = \frac{a}{15}$:

$a = \frac{20 \times 15}{9} = \frac{300}{9} = 33.33 \quad \text{(or } \frac{100}{3} \text{ as a fraction)}$

Step 2: Solve for $b$

Using the proportion $\frac{20}{9} = \frac{b}{12}$:

$b = \frac{20 \times 12}{9} = \frac{240}{9} = 26.67 \quad \text{(or } \frac{80}{3} \text{ as a fraction)}$

Final answers:

• $a = \frac{100}{3}$
• $b = \frac{80}{3}$

Would you like more details or have any questions?

Here are five related questions to explore:

1. How can you check if two triangles are similar?
2. What happens if the ratio of corresponding sides is not constant?
3. Can two triangles be similar if they have different angles?
4. How would you calculate the areas of these similar triangles?
5. Can this method be used for non-right triangles as well?

Tip: Always double-check your proportional relationships when working with similar triangles.