The mid-points of AB and AC of a `Delta ABC` are respectively X and Y. If BC + XY = 12 units, then the value of BC - XY is

Options

A. 6
B. 8
C. 4
D. 12

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Midpoint Theorem
The midpoint theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and is half its length. In this case, since X and Y are the midpoints of sides AB and AC respectively, we can write:
\[ XY = \frac{1}{2} BC \]

### Step 2: Set Up the Equation
From the problem, we know:
\[ BC + XY = 12 \]
We can substitute the expression for XY from the midpoint theorem into this equation:
\[ BC + \frac{1}{2} BC = 12 \]

### Step 3: Combine Like Terms
Now, we can combine the terms on the left side:
\[ \frac{3}{2} BC = 12 \]

### Step 4: Solve for BC
To find BC, we will multiply both sides by \( \frac{2}{3} \):
\[ BC = 12 \times \frac{2}{3} \]
\[ BC = 8 \text{ units} \]

### Step 5: Find XY
Now we can find XY using the relationship we established earlier:
\[ XY = \frac{1}{2} BC \]
Substituting the value of BC:
\[ XY = \frac{1}{2} \times 8 = 4 \text{ units} \]

### Step 6: Calculate BC - XY
Now we need to find the value of \( BC - XY \):
\[ BC - XY = 8 - 4 = 4 \text{ units} \]

### Final Answer
Thus, the value of \( BC - XY \) is:
\[ \boxed{4} \text{ units} \]

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