If `DeltaABC~DeltaQRP,(ar(DeltaABC))/(ar(DeltaPQR))=9/4`, AB=`18` cm and BC=`15` cm, then PR is equal to

Options

A. `10 cm`
B. `12 cm`
C. `20/3cm`
D. `8 cm`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the areas of similar triangles
Since triangle ABC is similar to triangle QRP, the ratio of their areas can be expressed in terms of the ratio of the squares of their corresponding sides.

### Step 2: Set up the ratio of the areas
We are given that:
\[ \frac{ar(\Delta ABC)}{ar(\Delta PQR)} = \frac{9}{4} \]
This means:
\[ \frac{AB^2}{QR^2} = \frac{BC^2}{RP^2} = \frac{AC^2}{QP^2} = \frac{9}{4} \]

### Step 3: Substitute the known values
We know that \( AB = 18 \, \text{cm} \) and \( BC = 15 \, \text{cm} \). We can use the ratio involving \( BC \) and \( RP \):
\[ \frac{15^2}{RP^2} = \frac{9}{4} \]

### Step 4: Solve for \( RP^2 \)
From the equation:
\[ \frac{15^2}{RP^2} = \frac{9}{4} \]
We can cross-multiply to find \( RP^2 \):
\[ 15^2 \cdot 4 = 9 \cdot RP^2 \]
\[ 225 \cdot 4 = 9 \cdot RP^2 \]
\[ 900 = 9 \cdot RP^2 \]
Now, divide both sides by 9:
\[ RP^2 = \frac{900}{9} = 100 \]

### Step 5: Take the square root to find \( RP \)
Now, we take the square root of both sides:
\[ RP = \sqrt{100} = 10 \, \text{cm} \]

### Conclusion
Thus, the length of \( PR \) is \( 10 \, \text{cm} \).

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