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

Answer

To solve the problem, we will use the Basic Proportionality Theorem (also known as Thales' theorem). Here’s a step-by-step solution:

### Step 1: Understand the given information
We have a triangle ABC with XY drawn parallel to the base BC. The line XY intersects AB at point X and AC at point Y. We are given:
- AB = 4BX
- YC = 2 cm

### Step 2: Set up the ratios using the Basic Proportionality Theorem
According to the Basic Proportionality Theorem, if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Therefore, we can write:
\[ \frac{AX}{BX} = \frac{AY}{YC} \]

### Step 3: Express AX in terms of BX
Since AB = 4BX, we can express AX as:
\[ AX = AB - BX = 4BX - BX = 3BX \]
Thus, we can substitute this into our ratio:
\[ \frac{3BX}{BX} = \frac{AY}{2} \]

### Step 4: Simplify the ratio
This simplifies to:
\[ 3 = \frac{AY}{2} \]

### Step 5: Cross-multiply to find AY
Cross-multiplying gives us:
\[ AY = 3 \times 2 = 6 \text{ cm} \]

### Conclusion
Thus, the length of AY is:
\[ AY = 6 \text{ cm} \]