If in `DeltaABC, DE||BC, AB = 7.5 cm, BD = 6 cm` and `DE = 2cm`, then the length of BC in cm is :

Options

A. 6
B. 8
C. 10
D. 10.5

Answer

To find the length of BC in triangle ABC given that DE is parallel to BC, we can use the properties of similar triangles. Here’s the step-by-step solution:

### Step 1: Identify the triangles and the given information
We have triangle ABC, where DE is parallel to BC. The lengths given are:
- AB = 7.5 cm
- BD = 6 cm
- DE = 2 cm

### Step 2: Use the properties of similar triangles
Since DE is parallel to BC, triangles ADE and ABC are similar by the Angle-Angle (AA) criterion. This means that the ratios of their corresponding sides are equal.

### Step 3: Set up the ratio of the sides
From the similarity of triangles, we can write the following ratio:
\[ \frac{AD}{AB} = \frac{DE}{BC} \]

### Step 4: Express AD in terms of AB and BD
Since AD = AB - BD, we can substitute this into the ratio:
\[ \frac{AB - BD}{AB} = \frac{DE}{BC} \]
Substituting the known values:
\[ \frac{7.5 - 6}{7.5} = \frac{2}{BC} \]

### Step 5: Simplify the left side
Calculating the left side:
\[ \frac{1.5}{7.5} = \frac{2}{BC} \]
This simplifies to:
\[ \frac{1}{5} = \frac{2}{BC} \]

### Step 6: Cross-multiply to solve for BC
Cross-multiplying gives us:
\[ 1 \cdot BC = 2 \cdot 5 \]
Thus:
\[ BC = 10 \text{ cm} \]

### Conclusion
The length of BC is **10 cm**.

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