Since triangles \( \triangle CDE \) and \( \triangle PQR \) are similar (denoted by \( \triangle CDE \sim \triangle PQR \)), the corresponding sides are in proportion.
We are given:
- \( CD = 9 , m \)
- \( EC = 15 , m \)
- \( PQ = 15 , m \)
To find the length of \( RP \), we can first find the total length of \( DE \) (which corresponds to \( QR \)) using the ratio of the sides.
The total length \( CE \) is: \[ CE = CD + EC = 9 , m + 15 , m = 24 , m \]
Now we set up the proportion using the lengths of the sides: \[ \frac{CD}{PQ} = \frac{CE}{RP} \]
Substitute in the known values: \[ \frac{9}{15} = \frac{24}{RP} \]
Cross-multiplying gives: \[ 9 \cdot RP = 15 \cdot 24 \] \[ 9 \cdot RP = 360 \] \[ RP = \frac{360}{9} \] \[ RP = 40 , m \]
By checking the response options, it seems we have an issue because none of the provided options (25 m, 9 m, 0.6 m, 30 m) include the calculated \( RP \) of 40 m.
If there are no transcription errors, the information provided might not correspond with the given options. You may want to double-check the question or the provided options. If you have to select the closest value among the provided options, the calculation suggests a mismatch. The computed correct answer is \( RP = 40 , m \).