To solve the problem, let's establish a coordinate system:
- Place village A at the origin: \( A(0, 0) \).
- Village B is 7 km south of A: \( B(0, -7) \).
- Village C is 9 km south of B: \( C(0, -16) \) (since \( -7 - 9 = -16 \)).
- 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.