As a meteor shower approached a space colony, Nova and Luna raced to activate a chain of orbital defense satellites. The control system displayed the satellites' coordinates and required the formation to satisfy several geometric checks before the shield could be enabled. They calculated the distances between neighboring satellites to verify spacing, determined the midpoints of key segments to position relay drones, and compared the slopes of communication links to confirm that critical alignments were parallel. After correcting one misplaced satellite based on their calculations, the system approved the formation. Moments before the meteors arrived, the protective shield activated and safely defended the colony.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Use the distance formula with horizontal change 6 and vertical change 8.
Two neighboring defense satellites are located at A(1, 2) and B(7, 10). Nova claims that the distance between them is 14 units because 6 + 8 = 14. Which statement correctly checks her claim?
Nova is correct because coordinate changes should be added.Nova is incorrect because the distance is 10 units.Nova is incorrect because the distance is 8 units.Nova is correct because 14 is the midpoint.Correct! The straight-line distance is 10 units, not 14 units.
The horizontal and vertical changes form the legs of a right triangle, so use the distance formula.
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Hint
Average the x-coordinates and the y-coordinates, then add the results.
A relay drone must be placed at the midpoint of the segment joining satellites C(4, 6) and D(14, 12). What is the sum of the midpoint's coordinates?
Correct! The midpoint is (9, 9), so the coordinate sum is 18.
Find each midpoint coordinate by averaging the matching endpoint coordinates.
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Hint
Find the slope of PQ, then test which endpoint gives RS the same slope.
One communication link joins P(2, 1) to Q(8, 5). A second link starts at R(5, 3) and must end at one of the following points so the two links are parallel. Which endpoint should Nova and Luna choose?
S(11, 5)S(9, 9)S(11, 7)S(8, 9)Correct! Both links have slope 2/3, so they are parallel.
Parallel links must have equal slopes. Calculate the slope from R to each possible endpoint.
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