To find the length between the points \((2, 5)\) and \((7, 3)\), we can use the distance formula, which is derived from the Pythagorean theorem. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case, we have:
- \((x_1, y_1) = (2, 5)\)
- \((x_2, y_2) = (7, 3)\)
Now plug in the values into the distance formula:
\[ d = \sqrt{(7 - 2)^2 + (3 - 5)^2} \]
Calculating each part:
- \(7 - 2 = 5\)
- \(3 - 5 = -2\)
Now square these values:
\[ d = \sqrt{5^2 + (-2)^2} \] \[ = \sqrt{25 + 4} \] \[ = \sqrt{29} \]
Now we can calculate \(\sqrt{29}\):
\[ \sqrt{29} \approx 5.385 \]
Rounding to the nearest hundredth, we get:
\[ d \approx 5.39 \]
The distance between the points \((2, 5)\) and \((7, 3)\) is approximately 5.39.
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