Question

you just found the length of the hypotenuse, c. thats the length of the line segment between point a and point b. label that on the coordinate plane.
the coordinate plane has points b(-1,5), c(7,5), a(7,-3). the equations are $a^2 + b^2 = c^2$, $8^2 + 8^2 = c^2$, $c = \sqrt{128}$

Explanation:

Step1: Simplify the square root

We know that \(128 = 64\times2\), and \(\sqrt{64\times2}=\sqrt{64}\times\sqrt{2}\) (by the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a\geq0,b\geq0\)). Since \(\sqrt{64} = 8\), we have \(\sqrt{128}=\sqrt{64\times2}=8\sqrt{2}\).

Step2: Calculate the numerical value (optional, but for clarity)

We know that \(\sqrt{2}\approx1.414\), so \(8\sqrt{2}\approx8\times1.414 = 11.312\). But the exact simplified form is \(8\sqrt{2}\).

Answer:

The length of the hypotenuse \(c\) is \(8\sqrt{2}\) (or approximately \(11.31\)).