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At At ch ge to 12. In the standard (x,y) coordinate plane, the point (2,1) is the midpoint of bar(CD). Point C has coordinates (6,8). What are the coordinates of point D ? F. (-2,-(7)/(2)) G. (-2,-6) H. (4,(9)/(2)) J. (10,10) K. (10,15)

At\newlineAt\newlinech\newlinege\newlineto\newline1212. In the standard (x,y) (x, y) coordinate plane, the point (2,1) (2,1) is the midpoint of CD‾ \overline{C D} . Point C C has coordinates (6,8) (6,8) . What are the coordinates of point D D ?\newlineF. (−2,−72) \left(-2,-\frac{7}{2}\right) \newlineG. (−2,−6) (-2,-6) \newlineH. (4,92) \left(4, \frac{9}{2}\right) \newlineJ. (10,10) (10,10) \newlineK. (2,1) (2,1) 00

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Full solution

Q. At\newlineAt\newlinech\newlinege\newlineto\newline1212. In the standard (x,y) (x, y) coordinate plane, the point (2,1) (2,1) is the midpoint of CD‾ \overline{C D} . Point C C has coordinates (6,8) (6,8) . What are the coordinates of point D D ?\newlineF. (−2,−72) \left(-2,-\frac{7}{2}\right) \newlineG. (−2,−6) (-2,-6) \newlineH. (4,92) \left(4, \frac{9}{2}\right) \newlineJ. (10,10) (10,10) \newlineK. (2,1) (2,1) 00
  1. Midpoint formula: To find the coordinates of point D, use the midpoint formula which states that the midpoint M(x,y) is calculated as M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right), where (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the endpoints.
  2. Equation setup: We know the midpoint is (2,1)(2,1) and one endpoint CC is (6,8)(6,8). Let's call the coordinates of DD (xd,yd)(x_d, y_d). We can set up the equations (6+xd)/2=2(6 + x_d)/2 = 2 and (8+yd)/2=1(8 + y_d)/2 = 1.
  3. Solve for xdx_d: Solve the first equation for xdx_d: (6+xd)/2=2(6 + x_d)/2 = 2. Multiply both sides by 22 to get 6+xd=46 + x_d = 4. Then subtract 66 from both sides to find xd=4−6x_d = 4 - 6.
  4. Calculate xdxd: xd=4−6xd = 4 - 6 gives xd=−2xd = -2. So the x-coordinate of point D is −2-2.
  5. Solve for ydy_d: Now solve the second equation for ydy_d: 8+yd2=1\frac{8 + y_d}{2} = 1. Multiply both sides by 22 to get 8+yd=28 + y_d = 2. Then subtract 88 from both sides to find yd=2−8y_d = 2 - 8.
  6. Calculate ydyd: yd=2−8yd = 2 - 8 gives yd=−6yd = -6. So the y-coordinate of point D is −6-6.
  7. Final coordinates: Therefore, the coordinates of point D are (−2,−6)(-2, -6), which corresponds to answer choice G.

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