Question

graph a right triangle with the two points forming the hypotenuse. using the sides, find the distance between the two points in simplest radical form. (-3,7) and (-1,9) click twice to draw a line. click a segment to erase it.

Explanation:

Step1: Find horizontal and vertical side - lengths

Let $(x_1,y_1)=(-3,7)$ and $(x_2,y_2)=(-1,9)$. The horizontal side - length (change in x) is $\Delta x=\vert x_2 - x_1\vert=\vert-1-(-3)\vert = 2$. The vertical side - length (change in y) is $\Delta y=\vert y_2 - y_1\vert=\vert9 - 7\vert=2$.

Step2: Apply the Pythagorean theorem

The distance $d$ between two points (the length of the hypotenuse) is given by the Pythagorean theorem $d=\sqrt{(\Delta x)^2+(\Delta y)^2}$. Substitute $\Delta x = 2$ and $\Delta y = 2$ into the formula: $d=\sqrt{2^2+2^2}=\sqrt{4 + 4}=\sqrt{8}$.

Step3: Simplify the radical

We can simplify $\sqrt{8}$ as $\sqrt{4\times2}=\sqrt{4}\times\sqrt{2}=2\sqrt{2}$.

Answer:

$2\sqrt{2}$