While mapping a newly discovered asteroid field, Nova and Luna recorded the coordinates of research beacons on a navigation grid. They measured the distances between neighboring beacons to confirm that communication links stayed within range, calculated midpoints to determine the best locations for additional relay devices, and compared the slopes of connecting paths to identify routes that were parallel or likely to intersect. Their calculations revealed a missing relay between two distant beacons. After updating the network and verifying every connection, they completed an accurate navigation map that future exploration missions could rely on.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Use the distance formula with horizontal change 6 and vertical change 8.
Two research beacons are located at A(1, 2) and B(7, 10). What is the distance between the beacons?
8 units10 units12 units14 unitsCorrect! The distance between the beacons is 10 units.
Square the horizontal and vertical changes, add them, and take the square root.
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Hint
Average the x-coordinates and the y-coordinates, then add the midpoint coordinates.
A missing relay must be placed at the midpoint between beacons C(4, 6) and D(12, 14). What is the sum of the relay's x-coordinate and y-coordinate?
Correct! The midpoint is (8, 10), and the coordinate sum is 18.
Find the midpoint first by averaging each pair of matching coordinates.
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Hint
Calculate each slope using change in y divided by change in x.
Nova and Luna compare two communication paths. Path PQ connects P(2, 1) to Q(8, 5), and path RS connects R(1, 4) to S(10, 10). What can they conclude about the paths?
The paths are parallel because both slopes are 2/3.The paths are perpendicular because their slopes are opposites.The paths are parallel because both have length 10 units.The paths are neither parallel nor perpendicular because their slopes are different.Correct! Both paths have slope 2/3, so they are parallel.
Compare the slopes of the two paths rather than only their positions or lengths.
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