On the Moon, Emek and Tara prepared three instruments to survey a newly discovered crater. Their coordinate map placed Instrument A at (0, 0) and Instrument B at (8, 0), with each coordinate unit representing one kilometer. For accurate measurements, Instrument C had to be exactly 5 kilometers from both A and B. Tara modeled the possible locations with the circles x^2 + y^2 = 25 and (x - 8)^2 + y^2 = 25. Subtracting the equations showed that both intersection points had x = 4. Substituting x = 4 into the first equation gave y^2 = 9, so y = 3 or y = -3. The two possible locations for Instrument C were therefore (4, 3) and (4, -3). Emek checked the terrain map and saw that the region below the x-axis contained a steep crater wall, while (4, 3) was on stable ground. They placed the third instrument at (4, 3), verified its 5-kilometer distance from both other instruments, and successfully began the lunar survey.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4
Hint
Use the standard equation of a circle with center (8, 0) and radius 5.
Instrument C must be 5 kilometers from Instrument B at (8, 0). Which equation represents all possible locations that are 5 kilometers from B?
x^2 + y^2 = 25(x - 5)^2 + y^2 = 8(x - 8)^2 + y^2 = 25(x + 8)^2 + y^2 = 25
Correct. A circle centered at (8, 0) with radius 5 has equation (x - 8)^2 + y^2 = 25.
Identify the circle's center and radius before writing its equation.
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Hint
The story gives the distances from C to A and B. Find the distance from A to B, then add all three side lengths.
Instrument C is placed at (4, 3), while Instrument A is at (0, 0) and Instrument B is at (8, 0). What is the perimeter, in kilometers, of triangle ABC?
Correct. The three side lengths are 5, 5, and 8 kilometers, giving a perimeter of 18 kilometers.
Use the 5-kilometer distances from C to both instruments and the coordinate distance from A to B.
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Hint
Consider what happens to y^2 when y changes from 3 to -3.
Why do the two possible locations (4, 3) and (4, -3) for Instrument C have the same distance from both A and B?
They are reflections across the x-axis, so changing the sign of y does not change the squared distance to A or B.They are both on the x-axis, so their vertical distances are zero.Their y-coordinates are equal because 3 = -3.Every point with x = 4 is exactly 5 kilometers from both A and B.
Correct. Squaring 3 or -3 gives the same value, so the reflected points have equal distances from A and B.
Compare the distance calculations for y = 3 and y = -3, paying attention to the squared vertical difference.
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Ay Araştırma Rotasını Tasarlamak
TR
ARC_G11_GM_02_STORY_02
Ay Araştırma Rotasını Tasarlamak
Designing the Lunar Survey Route
EN
ARC_G11_GM_02_STORY_02
Designing the Lunar Survey Route