Village B is 7km due south of village A and village C is 9km due south of B. Village D is 8km from A and is due east of B. Find the distance of D from B and C

To solve the problem, let's establish a coordinate system:

1. Place village A at the origin: \( A(0, 0) \).
2. Village B is 7 km south of A: \( B(0, -7) \).
3. Village C is 9 km south of B: \( C(0, -16) \) (since \( -7 - 9 = -16 \)).
4. Village D is 8 km from A and due east of B.

Since D is due east of B, it will have the same y-coordinate as B and an x-coordinate that is 8 km farther east of A. The x-coordinate of D is the distance from A which is \( 8 \) km:

- Therefore, the coordinates of D are \( D(8, -7) \).

Now we can find the distances from D to both B and C.

### Distance from D to B
Using the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Where:
- \( D(8, -7) \)
- \( B(0, -7) \)

Calculating the distance \( DB \):
\[
DB = \sqrt{(8 - 0)^2 + (-7 - (-7))^2} = \sqrt{8^2 + 0^2} = \sqrt{64} = 8 \text{ km}
\]

### Distance from D to C
Now calculate the distance from D to C using the same formula:
- \( C(0, -16) \)

Calculating the distance \( DC \):
\[
DC = \sqrt{(8 - 0)^2 + (-7 - (-16))^2} = \sqrt{8^2 + (-7 + 16)^2} = \sqrt{8^2 + 9^2}
\]
\[
= \sqrt{64 + 81} = \sqrt{145} \approx 12.04 \text{ km}
\]

### Summary
- Distance from D to B is \( 8 \) km.
- Distance from D to C is approximately \( 12.04 \) km.