Find the length of the following pair of points.

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Here is the solution for question 5 from the image.

5. Find the length of the following pair of point

The distance formula 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}

(1) $(-3,4)$ and $(-5,0)$

Identify the coordinates.
Let $(x_1, y_1) = (-3, 4)$ and $(x_2, y_2) = (-5, 0)$ .
Substitute the coordinates into the distance formula.
$D = \sqrt{(-5 - (-3))^2 + (0 - 4)^2}$ $D = \sqrt{(-5 + 3)^2 + (-4)^2}$ $D = \sqrt{(-2)^2 + 16}$ $D = \sqrt{4 + 16}$ $D = \sqrt{20}$
Simplify the radical.
$D = \sqrt{4 \times 5}$ $D = 2\sqrt{5}$
The final answer for 5(1) is $\boxed{2\sqrt{5}}$ .

(2) $(2,0)$ and $(3,4)$
Identify the coordinates.
Let $(x_1, y_1) = (2, 0)$ and $(x_2, y_2) = (3, 4)$ .
Substitute the coordinates into the distance formula.
$D = \sqrt{(3 - 2)^2 + (4 - 0)^2}$ $D = \sqrt{(1)^2 + (4)^2}$ $D = \sqrt{1 + 16}$ $D = \sqrt{17}$
The final answer for 5(2) is $\boxed{\sqrt{17}}$ .

(3) $(\frac{1}{2}, 1)$ and $(1, \frac{1}{2})$
Identify the coordinates.
Let $(x_1, y_1) = (\frac{1}{2}, 1)$ and $(x_2, y_2) = (1, \frac{1}{2})$ .
Substitute the coordinates into the distance formula.
$D = \sqrt{(1 - \frac{1}{2})^2 + (\frac{1}{2} - 1)^2}$ $D = \sqrt{(\frac{2}{2} - \frac{1}{2})^2 + (\frac{1}{2} - \frac{2}{2})^2}$ $D = \sqrt{(\frac{1}{2})^2 + (-\frac{1}{2})^2}$ $D = \sqrt{\frac{1}{4} + \frac{1}{4}}$ $D = \sqrt{\frac{2}{4}}$ $D = \sqrt{\frac{1}{2}}$
Rationalize the denominator.
$D = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$ $D = \frac{\sqrt{2}}{2}$
The final answer for 5(3) is $\boxed{\frac{\sqrt{2}}{2}}$ .

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