Jun 10, 2017

$2 \sqrt{10} \text{ }$ as an exact value

#### Explanation:

When 'reading' a graph in '2-space' (x and y axis) then you always read left to right on the x-axis

Let point 1 be ${P}_{1} \to \left({x}_{1} , {y}_{1}\right) = \left(- 1 , - 3\right)$
Let point 2 be ${P}_{2} \to \left({x}_{2} , {y}_{2}\right) = \left(1 , 3\right)$
Let the direct distance from ${P}_{1} \text{ to } {P}_{2}$ be $d$

The link between Pythagoras and the distance between two points is due to the triangle depicted in the graph below:

Thus we have: ${d}^{2} = {\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}$

$\implies d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$\implies d = \sqrt{{\left[1 - \left(- 1\right)\right]}^{2} + {\left[3 - \left(- 3\right)\right]}^{2}}$

$d = \sqrt{4 + 36} \text{ } = \sqrt{40}$

$d = \sqrt{{2}^{2} \times 10} \text{ "=2sqrt(10)" }$ as an exact value