Question

use the figure shown. what is the exact length of $overline{ac}$?

Explanation:

Step1: Count horizontal and vertical units

Count the number of horizontal and vertical units between points A and C. Assume each grid - square side is 1 unit. Horizontally, the distance from A to C is 8 units and vertically the distance is 2 units.

Step2: Apply the distance formula

The distance formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) in a coordinate - plane is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). In terms of grid units, if we consider the right - triangle formed with the line segment AC as the hypotenuse, by the Pythagorean theorem \(d=\sqrt{8^{2}+2^{2}}\).

Step3: Calculate the value

First, calculate \(8^{2}=64\) and \(2^{2}=4\). Then \(8^{2}+2^{2}=64 + 4=68\). So, \(d=\sqrt{68}\), which can be simplified as \(d = 2\sqrt{17}\) (since \(68=4\times17\) and \(\sqrt{68}=\sqrt{4\times17}=\sqrt{4}\times\sqrt{17}=2\sqrt{17}\)).

Answer:

\(2\sqrt{17}\)