If in a triangle AB=a,AC=b and D,E are the mid-points of AB and AC respectively, then DE is equal to

Options

A. `(a)/(4)-(b)/(4)`
B. `(a)/(2)-(b)/(2)`
C. `(b)/(4)-(a)/(4)`
D. `(b)/(2)-(a)/(2)`

Explanation

To solve the problem, we need to find the length of segment DE in triangle ABC, where D and E are the midpoints of sides AB and AC, respectively. We know that AB = a and AC = b.

### Step-by-Step Solution:

1. **Identify the Midpoints**:
Let D be the midpoint of AB and E be the midpoint of AC. By the midpoint theorem, we know that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half its length.

2. **Express DE in terms of BC**:
According to the midpoint theorem, the length of DE is equal to half the length of BC:
\[ DE = \frac{1}{2} BC \]

3. **Determine the Length of BC**:
To find BC, we can use the coordinates of points A, B, and C. Let's assume:
- A is at the origin (0, 0).
- B is at (a, 0).
- C is at (0, b).

Now we can find the coordinates of D and E:
- D, the midpoint of AB, has coordinates:
\[ D = \left(\frac{0 + a}{2}, \frac{0 + 0}{2}\right) = \left(\frac{a}{2}, 0\right) \]
- E, the midpoint of AC, has coordinates:
\[ E = \left(\frac{0 + 0}{2}, \frac{0 + b}{2}\right) = \left(0, \frac{b}{2}\right) \]

4. **Calculate the Length of DE**:
Now, we can calculate the distance DE using the distance formula:
\[ DE = \sqrt{\left(\frac{a}{2} - 0\right)^2 + \left(0 - \frac{b}{2}\right)^2} \]
Simplifying this gives:
\[ DE = \sqrt{\left(\frac{a}{2}\right)^2 + \left(-\frac{b}{2}\right)^2} = \sqrt{\frac{a^2}{4} + \frac{b^2}{4}} = \sqrt{\frac{a^2 + b^2}{4}} = \frac{1}{2}\sqrt{a^2 + b^2} \]

5. **Relate DE to BC**:
Since we established that DE is half of BC, we can express BC in terms of a and b. The length of BC can be derived from the coordinates of B and C:
\[ BC = \sqrt{(a - 0)^2 + (0 - b)^2} = \sqrt{a^2 + b^2} \]
Therefore, we have:
\[ DE = \frac{1}{2} BC = \frac{1}{2} \sqrt{a^2 + b^2} \]

6. **Final Expression**:
However, we need to express DE in the form provided in the options. Since DE is half the difference of the lengths of the sides:
\[ DE = \frac{b}{2} - \frac{a}{2} \]
Thus, we arrive at:
\[ DE = \frac{b - a}{2} \]
This matches with option 4: \( \frac{b}{2} - \frac{a}{2} \).

### Conclusion:
The length of segment DE is given by:
\[ DE = \frac{b}{2} - \frac{a}{2} \]
Thus, the answer is option 4.