To find the distance between two points \( E(x_1, y_1) \) and \( F(x_2, y_2) \), you use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In your expressions, it looks like you have points with coordinates. To clarify, let's assign coordinates to points \( E \) and \( F \). For instance, let's say:
- Point \( E \) has coordinates \((2, 5)\)
- Point \( F \) has coordinates \((6, 1)\)
Using the distance formula correctly:
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Substitute the coordinates into the formula: \[ d = \sqrt{(6 - 2)^2 + (1 - 5)^2} \]
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Calculate each part: \[ d = \sqrt{(4)^2 + (-4)^2} \] \[ = \sqrt{16 + 16} \] \[ = \sqrt{32} \] \[ = 4\sqrt{2} \]
Thus, the distance between points \( E(2, 5) \) and \( F(6, 1) \) is \( 4\sqrt{2} \).
Correction of Other Expressions
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The expression \( \sqrt{((-2) + 6)^2 + (5 + 1)^2} \) is incorrect in context. It does not represent the correct subtraction of coordinates required for the distance formula.
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The expression \( \sqrt{(2 - 6)^2 + (5 - 1)^2} \) is another form that correctly calculates the distance since \( x_2 - x_1 \) and \( y_2 - y_1 \) produce the same value by commutativity in squaring.
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The expressions \( \sqrt{(-6 - 2)^2 + (1 - (-5))^2} \) and others do not follow from the points provided, and should not be used unless you have points defined correctly to fit those expressions.
So, it's best to stick with the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) to ensure you are accurately calculating the distance between two points.
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