Question

alex is designing a car racing video game. alex wants to create a shortcut that allows a player to go straight from checkpoint a to the finish rather than going through checkpoint b before they reach the finish.

coordinate plane with checkpoint a at (-6,1), checkpoint b at (-4,-3), and finish at (6,-2)

coordinate values on the map are in units.
how much shorter is the shortcut straight to the finish than the path through checkpoint b?
round your final answer only to the nearest unit.

Explanation:

Step1: Find coordinates of points

Checkpoint A: \((-6, 1)\), Checkpoint B: \((-4, -3)\), Finish: \((6, -2)\)

Step2: Calculate distance from A to B (using distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\))

\(d_{AB} = \sqrt{(-4 - (-6))^2 + (-3 - 1)^2} = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.47\)

Step3: Calculate distance from B to Finish

\(d_{BF} = \sqrt{(6 - (-4))^2 + (-2 - (-3))^2} = \sqrt{(10)^2 + (1)^2} = \sqrt{100 + 1} = \sqrt{101} \approx 10.05\)

Step4: Calculate total path through B (\(d_{AB} + d_{BF}\))

\(d_{total} \approx 4.47 + 10.05 = 14.52\)

Step5: Calculate distance of shortcut (A to Finish)

\(d_{AF} = \sqrt{(6 - (-6))^2 + (-2 - 1)^2} = \sqrt{(12)^2 + (-3)^2} = \sqrt{144 + 9} = \sqrt{153} \approx 12.37\)

Step6: Find difference (\(d_{total} - d_{AF}\))

\(14.52 - 12.37 \approx 2\) (rounded to nearest unit)

Answer:

2