XY is drawn parallel to the base BC of a `DeltaABC` cutting AB at X and AC at Y. If AB= 4BX and YC = 2cm, then find AY.

Answer

To solve the problem step by step, we will use the properties of similar triangles.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We have triangle ABC with XY parallel to BC.
- It is given that AB = 4BX and YC = 2 cm.
- We need to find AY.

2. **Set Up the Relationship Using Similar Triangles:**
- Since XY is parallel to BC, triangles AXY and ABC are similar by the Basic Proportionality Theorem (also known as Thales' theorem).
- This gives us the proportional relationships:
\[ \frac{AX}{BX} = \frac{AY}{YC} \]

3. **Express AB in Terms of BX:**
- We know that AB = AX + BX.
- From the problem, we have AB = 4BX.
- Therefore, we can express AX as:
\[ AX = AB - BX = 4BX - BX = 3BX \]

4. **Substitute AX in the Proportional Relationship:**
- Now substituting AX in the proportion:
\[ \frac{3BX}{BX} = \frac{AY}{2} \]
- This simplifies to:
\[ 3 = \frac{AY}{2} \]

5. **Solve for AY:**
- Cross-multiplying gives:
\[ AY = 3 \times 2 = 6 \text{ cm} \]

### Final Answer:
AY = 6 cm

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