find the distance between the points $a$ and $b$ given below.\n(that is, find the length of the segment…

Explanation:

Step1: Determine horizontal and vertical distances

Counting the grid units, the horizontal change ($\Delta x$) is 2 units and the vertical change ($\Delta y$) is 5 units. $$ \Delta x = 2, \Delta y = 5 $$

Step2: Apply the Pythagorean theorem

The distance $d$ is the hypotenuse of a right triangle with legs $\Delta x$ and $\Delta y$. $$ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} $$

Step3: Substitute values and calculate

Substitute the identified values into the distance formula. $$ d = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} $$

Step4: Round to the nearest hundredth

Calculate the square root and round the result. $$ d \approx 5.38516... $$

Answer:

5.39