Let ABC be an acute angled triangle and CD be the altitude through C. If AB = 8 cm and CD = 6 cm, then the distance between the midpoints AD and BC is :

Options

A. 6
B. 4
C. 4.5
D. 5

Answer

To solve the problem, we need to find the distance between the midpoints of segments AD and BC in triangle ABC, where CD is the altitude from point C to side AB.

### Step-by-Step Solution:

1. **Identify the Triangle and Given Values:**
We have triangle ABC with:
- \( AB = 8 \, \text{cm} \)
- \( CD = 6 \, \text{cm} \) (the altitude from C to AB)

2. **Find the Coordinates of Points A, B, and C:**
We can place the triangle in a coordinate system for easier calculations:
- Let \( A(0, 0) \) and \( B(8, 0) \) (since AB = 8 cm lies on the x-axis).
- Point C will be directly above the midpoint of AB since CD is the altitude. The midpoint M of AB is at \( (4, 0) \).
- Since CD is vertical and has a length of 6 cm, the coordinates of point C will be \( C(4, 6) \).

3. **Calculate Midpoints of AD and BC:**
- **Midpoint of AD (let's call it M1):**
- A is at \( (0, 0) \) and D is at \( (4, 0) \) (the foot of the altitude).
- The midpoint M1 is calculated as:
\[ M1 = \left( \frac{0 + 4}{2}, \frac{0 + 0}{2} \right) = (2, 0) \]

- **Midpoint of BC (let's call it M2):**
- B is at \( (8, 0) \) and C is at \( (4, 6) \).
- The midpoint M2 is calculated as:
\[ M2 = \left( \frac{8 + 4}{2}, \frac{0 + 6}{2} \right) = \left( 6, 3 \right) \]

4. **Calculate the Distance Between M1 and M2:**
- The distance \( d \) between the points M1 and M2 can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Substituting the coordinates of M1 and M2:
\[ d = \sqrt{(6 - 2)^2 + (3 - 0)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{cm} \]

### Final Answer:
The distance between the midpoints AD and BC is \( 5 \, \text{cm} \).