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