To determine the locations of points R and S, we start with the coordinates of point Q at \((-9, -9)\). Since points R and S are each 88 units away from point Q, we can find their coordinates by applying the concept of distance in the coordinate plane.
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case, we want \(d = 88\) and \(Q = (-9, -9)\). If we let the coordinates of points R and S be \((x, y)\), we set up the equation:
\[ \sqrt{(x + 9)^2 + (y + 9)^2} = 88 \]
Squaring both sides to eliminate the square root gives:
\[ (x + 9)^2 + (y + 9)^2 = 88^2 \] \[ (x + 9)^2 + (y + 9)^2 = 7744 \]
This equation represents a circle with center \((-9, -9)\) and radius \(88\).
To find points R and S, we can consider some specific directions from point Q that are 88 units away:
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Move directly to the right (east): \[ R_1 = (-9 + 88, -9) = (79, -9) \]
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Move directly to the left (west): \[ R_2 = (-9 - 88, -9) = (-97, -9) \]
These give us two of the locations:
- Point R at \((79, -9)\)
- Point S at \((-97, -9)\)
We could find other possible coordinates by rotating around the point Q at different angles, but these provide two straightforward locations along the horizontal axis. Consequently, R and S could be:
- R at \((79, -9)\)
- S at \((-97, -9)\)
However, as mentioned, there are infinitely many positions for points R and S along the circumference of the circle defined above.
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