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To solve the given problem, we need to calculate the distances between points \(A(0, 9)\), \(B(-3, -7)\), and \(C(2, 19)\) using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
\]

### Step-by-Step Calculations:

#### 1. Distance \(d(A, B)\):
\[
d(A, B) = \sqrt{(-3 - 0)^2 + (-7 - 9)^2}
= \sqrt{(-3)^2 + (-16)^2}
= \sqrt{9 + 256}
= \sqrt{265}.
\]

#### 2. Distance \(d(B, C)\):
\[
d(B, C) = \sqrt{(2 - (-3))^2 + (19 - (-7))^2}
= \sqrt{(2 + 3)^2 + (19 + 7)^2}
= \sqrt{5^2 + 26^2}
= \sqrt{25 + 676}
= \sqrt{701}.
\]

#### 3. Distance \(d(A, C)\):
\[
d(A, C) = \sqrt{(2 - 0)^2 + (19 - 9)^2}
= \sqrt{2^2 + 10^2}
= \sqrt{4 + 100}
= \sqrt{104}.
\]

### Final Answers:
\[
d(A, B) = \sqrt{265}, \quad d(B, C) = \sqrt{701}, \quad d(A, C) = \sqrt{104}.
\]

The correct option matches these calculations. The second option from the provided choices is correct.

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Do you have any questions or want me to explain further? Here are some related questions to deepen understanding:

1. How is the distance formula derived from the Pythagorean theorem?
2. Can we verify the calculations using a different method or tool?
3. How does the geometry of the points (e.g., forming a triangle) affect these distances?
4. What would be the midpoint coordinates for each pair of points?
5. How would these calculations differ in three-dimensional space?

**Tip:** Always double-check square and square-root calculations to ensure accuracy.

Use the following coordinates to find d(A,B), d(B,C), and d(A,C): A(0,9), B(-3,-7), C(2,19).

Learn how to calculate the distances between points A(0,9), B(-3,-7), and C(2,19) using the distance formula. This problem covers key concepts in coordinate geometry, including applying the distance formula derived from the Pythagorean theorem.

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