If is given that `DeltaABC~DeltaYZX` and Ar `DeltaABC: Ar DeltaXYZ=256:25`. If AB = 12 cm, BC = 10cm, CA = 15cm, then what is the value of YZ (in cm)?

Options

A. `(10)/(3)`
B. `(75)/(16)`
C. `(25)/(8)`
D. `(15)/(4)`

Answer

To solve the problem, we need to find the length of side YZ in triangle YZX, given that triangle ABC is similar to triangle YZX and the ratio of their areas is 256:25. We also know the lengths of the sides of triangle ABC.

### Step-by-Step Solution:

1. **Understanding Similar Triangles**:
Since triangles ABC and YZX are similar, the ratio of their corresponding sides is equal to the square root of the ratio of their areas.

2. **Area Ratio**:
The area ratio is given as:
\[ \text{Area} \, \Delta ABC : \text{Area} \, \Delta YZX = 256 : 25 \]

3. **Finding the Ratio of Sides**:
To find the ratio of the sides, we take the square root of the area ratio:
\[ \frac{\text{Area} \, \Delta ABC}{\text{Area} \, \Delta YZX} = \left(\frac{AB}{YZ}\right)^2 \]
Therefore,
\[ \frac{AB}{YZ} = \sqrt{\frac{256}{25}} = \frac{16}{5} \]

4. **Given Length of AB**:
We know that \( AB = 12 \, \text{cm} \). So, we can set up the equation:
\[ \frac{12}{YZ} = \frac{16}{5} \]

5. **Cross Multiplying**:
Cross multiplying gives us:
\[ 12 \cdot 5 = 16 \cdot YZ \]
Simplifying this, we get:
\[ 60 = 16 \cdot YZ \]

6. **Solving for YZ**:
Now, divide both sides by 16 to find YZ:
\[ YZ = \frac{60}{16} = \frac{15}{4} \, \text{cm} \]

### Final Answer:
The value of YZ is \( \frac{15}{4} \, \text{cm} \).

---