To find the distance between the points \(\((-10, 3)\)\) and \(\((-4, 1)\)\), we use the distance formula:
\(\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]\)
Let's write out the coordinates:
- \(\( (x_1, y_1) = (-10, 3) \)\)
- \(\( (x_2, y_2) = (-4, 1) \)\)
First, calculate the differences in the x and y coordinates:
\(\[ \Delta x = x_2 - x_1 = -4 - (-10) = -4 + 10 = 6 \]\)
\(\[ \Delta y = y_2 - y_1 = 1 - 3 = -2 \]\)
Now, substitute \(\(\Delta x\)\) and \(\(\Delta y\)\) back into the distance formula:
\(\[ d = \sqrt{(6)^2 + (-2)^2} \]\)
\(\[ d = \sqrt{36 + 4} \]\)
\(\[ d = \sqrt{40} \]\)
The distance can be simplified further. Factor 40 into its prime factors:
\(\[ 40 = 4 \times 10 = 4 \times 2 \times 5 = 2^2 \times 10 \]\)
Using the property of square roots:
\(\[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10} \]\)
Therefore, the distance between the points \(\((-10, 3)\)\) and \(\((-4, 1)\)\) is:
\(\[ d = 2\sqrt{10} \]\)
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