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.
$(-4,7)$ and $(2,9)$
click twice to draw a line. click a segment to erase it.

Explanation:

Step1: Calculate the horizontal and vertical distances

The horizontal distance \( \Delta x\) between \((-4,7)\) and \((2,9)\) is \(x_2 - x_1=2-(-4)=6\).
The vertical distance \( \Delta y\) is \(y_2 - y_1 = 9 - 7=2\).

Step2: Apply the Pythagorean theorem

Let \(d\) be the distance between the two points. By the Pythagorean theorem \(d=\sqrt{(\Delta x)^2+(\Delta y)^2}\).
Substitute \(\Delta x = 6\) and \(\Delta y=2\) into the formula: \(d=\sqrt{6^{2}+2^{2}}=\sqrt{36 + 4}=\sqrt{40}\).

Step3: Simplify the radical

Simplify \(\sqrt{40}\): \(\sqrt{40}=\sqrt{4\times10}=2\sqrt{10}\).

Answer:

The distance between the two points \((-4,7)\) and \((2,9)\) is \(2\sqrt{10}\).