To solve this problem:

1. Prove that the two triangles are similar:

   • Two triangles are similar if they have equal angles or their corresponding sides are proportional.
   • From the diagram:
     • Both triangles share the angle $\angle R$ at the bottom.
     • Both triangles have a right angle (marked $90^\circ$).
     • Hence, the triangles are similar by AA (Angle-Angle) similarity criterion.

2. Find the value of $z$:

   • Use the proportionality of corresponding sides of similar triangles.
   • The larger triangle is $\triangle RPQ$ and the smaller triangle is $\triangle RST$.
   • The corresponding sides are proportional: $\frac{RS}{RP} = \frac{RT}{RQ}$
   • Substitute known values:
     • $RS = 4 \, \text{cm}$
     • $RT = 5 \, \text{cm}$
     • $RP = z$
     • $RQ = z + 4$
   • Using the proportion: $\frac{4}{z} = \frac{5}{z + 4}$
   • Cross-multiply: $4(z + 4) = 5z$
   • Expand and solve for $z$: $4z + 16 = 5z$ $16 = z$

Thus, the value of $z$ is 16 cm.

Would you like a breakdown of any step or additional related questions?

Related Questions:

1. What are the conditions for triangles to be similar?
2. Can the sides of a right triangle always be used for similarity proof?
3. How do you calculate proportions for non-right triangles?
4. What is the importance of AA similarity in geometry?
5. How would the solution change if additional side lengths were unknown?

Tip:

When working with similar triangles, always identify corresponding angles or sides first to establish proportionality!