If `triangleABC ~ triangleQRP, ("Area "(triangleABC))/("Area"(trianglePQR))=9/4,AB=18cm, BC=15cm`, then find the length of PR.

Answer

To solve the problem step by step, we will use the properties of similar triangles and the relationship between the areas of similar triangles.

### Step 1: Understand the relationship between the areas of similar triangles
Since triangle ABC is similar to triangle QRP, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

### Step 2: Set up the ratio of the areas
We are given that:
\[ \frac{\text{Area of triangle ABC}}{\text{Area of triangle QRP}} = \frac{9}{4} \]
This means:
\[ \frac{AB^2}{QR^2} = \frac{9}{4} \]

### Step 3: Find the ratio of the sides
Let the ratio of the sides be \( k \). Since the areas are in the ratio of \( 9:4 \), we can express this as:
\[ k^2 = \frac{9}{4} \]
Taking the square root of both sides gives:
\[ k = \frac{3}{2} \]

### Step 4: Use the side lengths to find PR
We know the lengths of sides AB and BC:
- \( AB = 18 \, \text{cm} \)
- \( BC = 15 \, \text{cm} \)

Since \( AB \) corresponds to \( QR \) and \( BC \) corresponds to \( RP \), we can write:
\[ \frac{AB}{QR} = \frac{3}{2} \quad \text{and} \quad \frac{BC}{RP} = \frac{3}{2} \]

### Step 5: Calculate QR using AB
Using the ratio for \( AB \):
\[ \frac{18}{QR} = \frac{3}{2} \]
Cross-multiplying gives:
\[ 3 \cdot QR = 2 \cdot 18 \]
\[ 3 \cdot QR = 36 \implies QR = \frac{36}{3} = 12 \, \text{cm} \]

### Step 6: Calculate PR using BC
Now we can use the ratio for \( BC \):
\[ \frac{15}{RP} = \frac{3}{2} \]
Cross-multiplying gives:
\[ 3 \cdot RP = 2 \cdot 15 \]
\[ 3 \cdot RP = 30 \implies RP = \frac{30}{3} = 10 \, \text{cm} \]

### Final Answer
Thus, the length of \( PR \) is:
\[ \boxed{10 \, \text{cm}} \]