What is the distance between the points (2,7) and (6,-2)?

Understand the Problem

The question is asking for the calculation of the distance between two points in a Cartesian plane. We will use the distance formula, which is derived from the Pythagorean theorem, to find the distance between the points (2,7) and (6,-2).

Answer

The distance is $ d = \sqrt{97} $ or approximately $ 9.85 $.

Steps to Solve

Identify the coordinates of the points

The points given are ( (x_1, y_1) = (2, 7) ) and ( (x_2, y_2) = (6, -2) ).

Use the distance formula

The distance ( d ) between two points in a Cartesian plane is given by the formula: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Substitute the coordinates into the formula

Now, substitute the coordinates of the points into the distance formula: $$ d = \sqrt{(6 - 2)^2 + (-2 - 7)^2} $$

Calculate the differences and squares

Calculate each difference:

( x_2 - x_1 = 6 - 2 = 4 )
( y_2 - y_1 = -2 - 7 = -9 )

Now substitute these values: $$ d = \sqrt{(4)^2 + (-9)^2} $$

Calculate the squares

Now we find the squares of 4 and -9: $$ d = \sqrt{16 + 81} $$

Add the values and take the square root

Finally, sum the results and find the square root: $$ d = \sqrt{97} $$

Final approximation of the distance

The final distance can be approximated as: $$ d \approx 9.85 $$

The distance between the points ( (2, 7) ) and ( (6, -2) ) is ( d = \sqrt{97} ) or approximately ( 9.85 ).

More Information

The distance formula is essential for finding the straight-line distance between two points in a plane. The square root of 97 does not simplify further, so the answer remains in its radical form.

Tips

Mixing up the coordinates when substituting into the distance formula. Always double-check your values for ( x_1, y_1, x_2, y_2 ).
Forgetting to square the differences before adding them. This is vital in applying the Pythagorean theorem correctly.

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