Answer :
Final answer:
To find the distance between points A and B, we use the distance formula. The coordinates for two points 7 units away from point C can be found by using the distance formula again and solving for the coordinates.
Explanation:
a.
To find the distance between two points, we use the distance formula:
d = √[(x2 - x1)² + (y2 - y1)²]
In this case, the coordinates of point A are (x1, y1) and the coordinates of point B are (x2, y2).
Using the given coordinates of A(-2, -3) and B(8, -9), we can substitute the values into the formula:
d = √[(8 - (-2))² + (-9 - (-3))²]
d = √[10² + (-6)²]
d = √(100 + 36)
d = √136
So, the distance between points A and B is √136 units.
b.
Given that point C has the coordinates (6, -1), we can find two points that are 7 units away from C by using the distance formula again.
Let's call the coordinates of the first point (x, y). Using the distance formula, we have:
d = √[(x - 6)² + (y - (-1))²] = 7
Simplifying this equation gives us:
(x - 6)² + (y + 1)² = 49
We can choose different values for x and solve for y to find various points that are 7 units away from C. For example, if we let x = 0, we get:
(-6)² + (y + 1)² = 49
36 + (y + 1)² = 49
(y + 1)² = 13
y + 1 = √13 or y + 1 = -√13
y = √13 - 1 or y = -√13 - 1
Therefore, two points that are 7 units away from C are (0, √13 - 1) and (0, -√13 - 1).
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