To find the coordinates of point $D$ such that $AB = CD$, let's proceed step by step.

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1. Calculate the length of $AB$:

Using the distance formula:

$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

where $A(2, -7)$ and $B(1, 3)$. Substituting these values:

$AB = \sqrt{(1 - 2)^2 + (3 - (-7))^2}$ $AB = \sqrt{(-1)^2 + (10)^2}$ $AB = \sqrt{1 + 100} = \sqrt{101}$

So, $AB = \sqrt{101}$.

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2. Let $D(x, y)$, and calculate $CD$ using the same formula:

We need $CD = AB = \sqrt{101}$. With $C(4, -1)$ and $D(x, y)$:

$CD = \sqrt{(x - 4)^2 + (y - (-1))^2} = \sqrt{(x - 4)^2 + (y + 1)^2}$

Set $CD = \sqrt{101}$: $\sqrt{(x - 4)^2 + (y + 1)^2} = \sqrt{101}$

Squaring both sides: $(x - 4)^2 + (y + 1)^2 = 101$

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3. Solve for $D(x, y)$:

The above equation represents a circle centered at $C(4, -1)$ with radius $\sqrt{101}$. To find possible coordinates for $D$, we select specific constraints for $D$. For simplicity, assume $D$ lies horizontally or vertically from $C$.

Case 1: $D$ lies horizontally (same $y$-coordinate as $C$):

If $y = -1$, substitute into the equation: $(x - 4)^2 + (-1 + 1)^2 = 101$ $(x - 4)^2 = 101$ $x - 4 = \pm \sqrt{101}$ $x = 4 \pm \sqrt{101}$

This gives two points: $D(4 + \sqrt{101}, -1) \quad \text{and} \quad D(4 - \sqrt{101}, -1)$

Case 2: $D$ lies vertically (same $x$-coordinate as $C$):

If $x = 4$, substitute into the equation: $(4 - 4)^2 + (y + 1)^2 = 101$ $(y + 1)^2 = 101$ $y + 1 = \pm \sqrt{101}$ $y = -1 \pm \sqrt{101}$

This gives two points: $D(4, -1 + \sqrt{101}) \quad \text{and} \quad D(4, -1 - \sqrt{101})$

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4. Final Answer:

The coordinates of $D$ can be one of the following pairs:

1. $(4 + \sqrt{101}, -1)$
2. $(4 - \sqrt{101}, -1)$
3. $(4, -1 + \sqrt{101})$
4. $(4, -1 - \sqrt{101})$

Which option is correct depends on additional constraints or the context of the problem.

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Do you need further clarifications or a specific visualization?

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Related Questions:

1. How do you determine the equation of a circle given two points?
2. What are the properties of distance in a coordinate plane?
3. How can you generalize the solution when no assumptions are made about the positioning of $D$?
4. What does it mean geometrically for $AB = CD$?
5. Can you apply this problem to find the equation of a line through $C$ and $D$?

Tip:

When solving for unknown points geometrically, consider simplifying assumptions (horizontal or vertical alignment) to reduce complexity.