Math Problem Statement

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).

Solution

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

d=(x2x1)2+(y2y1)2.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)^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. --- 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.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Distance Formula

Formulas

Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 9-12