In `Delta` ABC, DE || AC, D and E are two points on AB and CB respectively, If AB=10 cm and AD =4 cm, then BE : CE is

Options

A. `2:3`
B. `2 : 5`
C. `5 : 2`
D. `3 : 2`

Answer

To solve the problem, we will use the properties of similar triangles. Given that DE is parallel to AC in triangle ABC, we can apply the Basic Proportionality Theorem (also known as Thales's theorem), which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- In triangle ABC, we have:
- AB = 10 cm
- AD = 4 cm
- Since DE is parallel to AC, we can denote:
- BD = AB - AD = 10 cm - 4 cm = 6 cm.

2. **Set Up the Proportionality:**
- According to the Basic Proportionality Theorem, we have:
\[ \frac{AD}{DB} = \frac{AE}{EC} \]
- Here, AD = 4 cm and DB = 6 cm.

3. **Calculate the Ratio:**
- Substitute the known values into the proportion:
\[ \frac{4}{6} = \frac{AE}{EC} \]
- Simplifying the left side:
\[ \frac{2}{3} = \frac{AE}{EC} \]

4. **Express the Ratio of BE to CE:**
- Since AE + EC = AC and we know that AE = 2x and EC = 3x for some x (from the ratio):
- Thus, AC = AE + EC = 2x + 3x = 5x.
- Now, we can express BE in terms of EC:
\[ BE = AB - AE = 10 - 2x \]
- Therefore, the ratio of BE to CE is:
\[ \frac{BE}{CE} = \frac{10 - 2x}{3x} \]

5. **Find the Specific Ratio:**
- To find the ratio of BE to CE, we can set the total lengths in terms of a common variable. We already know from the previous step that:
\[ \frac{BE}{CE} = \frac{10 - 2x}{3x} \]
- Since we have the ratio of AE to EC as 2:3, we can directly conclude that:
\[ BE : CE = 3 : 2 \]

### Final Answer:
Thus, the ratio BE : CE is **3 : 2**.