The areas of two similar `triangleABC and trianglePQR` are 64 sq. cm and 121 sq. cm. repsectively. If QR= 15.4 cm, find BC.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the relationship between the areas of similar triangles
The areas of two similar triangles are proportional to the squares of their corresponding sides. This means that if we have two similar triangles, the ratio of their areas can be expressed as:

\[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle PQR} = \frac{(BC)^2}{(QR)^2} \]

### Step 2: Write down the known areas and side lengths
From the problem, we know:
- Area of triangle ABC = 64 sq. cm
- Area of triangle PQR = 121 sq. cm
- Length of side QR = 15.4 cm

### Step 3: Set up the equation using the areas
Using the relationship established in Step 1, we can set up the equation:

\[ \frac{64}{121} = \frac{(BC)^2}{(15.4)^2} \]

### Step 4: Calculate \( (15.4)^2 \)
First, we need to calculate \( (15.4)^2 \):

\[ (15.4)^2 = 237.16 \]

### Step 5: Substitute the value into the equation
Now substitute \( (15.4)^2 \) into the equation:

\[ \frac{64}{121} = \frac{(BC)^2}{237.16} \]

### Step 6: Cross-multiply to solve for \( (BC)^2 \)
Cross-multiplying gives us:

\[ 64 \times 237.16 = 121 \times (BC)^2 \]

Calculating \( 64 \times 237.16 \):

\[ 64 \times 237.16 = 15178.24 \]

So we have:

\[ 15178.24 = 121 \times (BC)^2 \]

### Step 7: Solve for \( (BC)^2 \)
Now divide both sides by 121:

\[ (BC)^2 = \frac{15178.24}{121} \]

Calculating this gives:

\[ (BC)^2 = 125.24 \]

### Step 8: Take the square root to find BC
Now take the square root of both sides to find BC:

\[ BC = \sqrt{125.24} \approx 11.2 \text{ cm} \]

### Final Answer
Thus, the length of side BC is approximately **11.2 cm**.

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