If `DeltaABC~DeltaPQR,(ar(DeltaABC))/(ar(DeltaPQR))=16/9`,AB=2cm and AC= 12 cm, then find the value of PR.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the areas of similar triangles
Given that triangles \( \Delta ABC \) and \( \Delta PQR \) are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

### Step 2: Set up the equation for the areas
We know that:
\[ \frac{ar(\Delta ABC)}{ar(\Delta PQR)} = \left(\frac{AB}{PQ}\right)^2 = \left(\frac{AC}{PR}\right)^2 \]
Given that:
\[ \frac{ar(\Delta ABC)}{ar(\Delta PQR)} = \frac{16}{9} \]

### Step 3: Relate the sides using the area ratio
From the area ratio, we can write:
\[ \left(\frac{AC}{PR}\right)^2 = \frac{16}{9} \]

### Step 4: Take the square root of both sides
Taking the square root of both sides gives us:
\[ \frac{AC}{PR} = \frac{4}{3} \]

### Step 5: Substitute the known value of AC
We know that \( AC = 12 \, cm \). Substituting this value into the equation gives:
\[ \frac{12}{PR} = \frac{4}{3} \]

### Step 6: Cross-multiply to solve for PR
Cross-multiplying gives us:
\[ 12 \cdot 3 = 4 \cdot PR \]
\[ 36 = 4 \cdot PR \]

### Step 7: Isolate PR
Now, divide both sides by 4:
\[ PR = \frac{36}{4} = 9 \, cm \]

### Final Answer
Thus, the value of \( PR \) is \( 9 \, cm \).

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