Mid-points of the sides AB and AC of a `DeltaABC` are (3, 5) and (-3, -3) respectively, then the length of the side BC is

Options

A. 10 unit
B. 20 unit
C. 15 unit
D. 30 unit

Answer

To find the length of side BC of triangle ABC, given the midpoints of sides AB and AC, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Midpoints**:
- Let the midpoints of sides AB and AC be M(3, 5) and N(-3, -3) respectively.

2. **Use the Midpoint Formula**:
- The coordinates of the midpoints M and N can be expressed in terms of the coordinates of points A, B, and C.
- If A(x1, y1), B(x2, y2), and C(x3, y3), then:
- M = ((x1 + x2)/2, (y1 + y2)/2) = (3, 5)
- N = ((x1 + x3)/2, (y1 + y3)/2) = (-3, -3)

3. **Set Up Equations**:
- From M(3, 5):
- (x1 + x2)/2 = 3 → x1 + x2 = 6 (Equation 1)
- (y1 + y2)/2 = 5 → y1 + y2 = 10 (Equation 2)
- From N(-3, -3):
- (x1 + x3)/2 = -3 → x1 + x3 = -6 (Equation 3)
- (y1 + y3)/2 = -3 → y1 + y3 = -6 (Equation 4)

4. **Express x2 and y2 in terms of x1 and y1**:
- From Equation 1: x2 = 6 - x1
- From Equation 2: y2 = 10 - y1

5. **Express x3 and y3 in terms of x1 and y1**:
- From Equation 3: x3 = -6 - x1
- From Equation 4: y3 = -6 - y1

6. **Find the Length of BC**:
- The length of side BC can be calculated using the distance formula:
\[ BC = \sqrt{(x3 - x2)^2 + (y3 - y2)^2} \]
- Substitute x2 and y2:
\[ BC = \sqrt{((-6 - x1) - (6 - x1))^2 + ((-6 - y1) - (10 - y1))^2} \]
- Simplifying:
\[ BC = \sqrt{(-12)^2 + (-16)^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \]

### Final Answer:
The length of side BC is 20.