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in exercises 15 - 18, the endpoints of $overline{cd}$ are given. find t…

Question

in exercises 15 - 18, the endpoints of $overline{cd}$ are given. find the coordinates of the midpoint $m$. (see example 3.)

  1. $c(3, - 5)$ and $d(7,9)$
  2. $c(-4,7)$ and $d(0, - 3)$
  3. $c(-2,0)$ and $d(4,9)$
  4. $c(-8, - 6)$ and $d(-4,10)$

in exercises 19 - 22, the midpoint $m$ and one endpoint of $overline{gh}$ are given. find the coordinates of the other endpoint. (see example 3.)

  1. $g(5, - 6)$ and $m(4,3)$ 20. $h(-3,7)$ and $m(-2,5)$
  2. $h(-2,9)$ and $m(8,0)$
  3. $g(-4,1)$ and $m(-\frac{13}{2}, - 6)$

Explanation:

Response
  1. For Exercises 15 - 18 (finding the mid - point):
  • The mid - point formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
  • Exercise 15:
  • Given \(C(3,-5)\) and \(D(7,9)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the mid - point
  • \(x=\frac{3 + 7}{2}=\frac{10}{2}=5\)
  • ## Step2: Calculate the y - coordinate of the mid - point
  • \(y=\frac{-5 + 9}{2}=\frac{4}{2}=2\)
  • # Answer:
  • \(M(5,2)\)
  • Exercise 16:
  • Given \(C(-4,7)\) and \(D(0,-3)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the mid - point
  • \(x=\frac{-4+0}{2}=\frac{-4}{2}=-2\)
  • ## Step2: Calculate the y - coordinate of the mid - point
  • \(y=\frac{7+( - 3)}{2}=\frac{4}{2}=2\)
  • # Answer:
  • \(M(-2,2)\)
  • Exercise 17:
  • Given \(C(-2,0)\) and \(D(4,9)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the mid - point
  • \(x=\frac{-2 + 4}{2}=\frac{2}{2}=1\)
  • ## Step2: Calculate the y - coordinate of the mid - point
  • \(y=\frac{0 + 9}{2}=4.5\)
  • # Answer:
  • \(M(1,4.5)\)
  • Exercise 18:
  • Given \(C(-8,-6)\) and \(D(-4,10)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the mid - point
  • \(x=\frac{-8+( - 4)}{2}=\frac{-12}{2}=-6\)
  • ## Step2: Calculate the y - coordinate of the mid - point
  • \(y=\frac{-6 + 10}{2}=\frac{4}{2}=2\)
  • # Answer:
  • \(M(-6,2)\)
  1. For Exercises 19 - 22 (finding the other endpoint):
  • Let one endpoint be \((x_1,y_1)\), the mid - point be \((x_m,y_m)\), and the other endpoint be \((x_2,y_2)\). Then \(x_m=\frac{x_1 + x_2}{2}\) and \(y_m=\frac{y_1 + y_2}{2}\), so \(x_2 = 2x_m-x_1\) and \(y_2 = 2y_m-y_1\).
  • Exercise 19:
  • Given \(G(5,-6)\) and \(M(4,3)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the other endpoint
  • \(x=2\times4 - 5=8 - 5 = 3\)
  • ## Step2: Calculate the y - coordinate of the other endpoint
  • \(y=2\times3-( - 6)=6 + 6=12\)
  • # Answer:
  • \(H(3,12)\)
  • Exercise 20:
  • Given \(H(-3,7)\) and \(M(-2,5)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the other endpoint
  • \(x=2\times(-2)-(-3)=-4 + 3=-1\)
  • ## Step2: Calculate the y - coordinate of the other endpoint
  • \(y=2\times5 - 7=10 - 7 = 3\)
  • # Answer:
  • \(G(-1,3)\)
  • Exercise 21:
  • Given \(H(-2,9)\) and \(M(8,0)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the other endpoint
  • \(x=2\times8-(-2)=16 + 2=18\)
  • ## Step2: Calculate the y - coordinate of the other endpoint
  • \(y=2\times0 - 9=-9\)
  • # Answer:
  • \(G(18,-9)\)
  • Exercise 22:
  • Given \(G(-4,1)\) and \(M(-\frac{13}{2},-6)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the other endpoint
  • \(x=2\times(-\frac{13}{2})-(-4)=-13 + 4=-9\)
  • ## Step2: Calculate the y - coordinate of the other endpoint
  • \(y=2\times(-6)-1=-12 - 1=-13\)
  • # Answer:
  • \(H(-9,-13)\)

Answer:

  1. For Exercises 15 - 18 (finding the mid - point):
  • The mid - point formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
  • Exercise 15:
  • Given \(C(3,-5)\) and \(D(7,9)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the mid - point
  • \(x=\frac{3 + 7}{2}=\frac{10}{2}=5\)
  • ## Step2: Calculate the y - coordinate of the mid - point
  • \(y=\frac{-5 + 9}{2}=\frac{4}{2}=2\)
  • # Answer:
  • \(M(5,2)\)
  • Exercise 16:
  • Given \(C(-4,7)\) and \(D(0,-3)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the mid - point
  • \(x=\frac{-4+0}{2}=\frac{-4}{2}=-2\)
  • ## Step2: Calculate the y - coordinate of the mid - point
  • \(y=\frac{7+( - 3)}{2}=\frac{4}{2}=2\)
  • # Answer:
  • \(M(-2,2)\)
  • Exercise 17:
  • Given \(C(-2,0)\) and \(D(4,9)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the mid - point
  • \(x=\frac{-2 + 4}{2}=\frac{2}{2}=1\)
  • ## Step2: Calculate the y - coordinate of the mid - point
  • \(y=\frac{0 + 9}{2}=4.5\)
  • # Answer:
  • \(M(1,4.5)\)
  • Exercise 18:
  • Given \(C(-8,-6)\) and \(D(-4,10)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the mid - point
  • \(x=\frac{-8+( - 4)}{2}=\frac{-12}{2}=-6\)
  • ## Step2: Calculate the y - coordinate of the mid - point
  • \(y=\frac{-6 + 10}{2}=\frac{4}{2}=2\)
  • # Answer:
  • \(M(-6,2)\)
  1. For Exercises 19 - 22 (finding the other endpoint):
  • Let one endpoint be \((x_1,y_1)\), the mid - point be \((x_m,y_m)\), and the other endpoint be \((x_2,y_2)\). Then \(x_m=\frac{x_1 + x_2}{2}\) and \(y_m=\frac{y_1 + y_2}{2}\), so \(x_2 = 2x_m-x_1\) and \(y_2 = 2y_m-y_1\).
  • Exercise 19:
  • Given \(G(5,-6)\) and \(M(4,3)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the other endpoint
  • \(x=2\times4 - 5=8 - 5 = 3\)
  • ## Step2: Calculate the y - coordinate of the other endpoint
  • \(y=2\times3-( - 6)=6 + 6=12\)
  • # Answer:
  • \(H(3,12)\)
  • Exercise 20:
  • Given \(H(-3,7)\) and \(M(-2,5)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the other endpoint
  • \(x=2\times(-2)-(-3)=-4 + 3=-1\)
  • ## Step2: Calculate the y - coordinate of the other endpoint
  • \(y=2\times5 - 7=10 - 7 = 3\)
  • # Answer:
  • \(G(-1,3)\)
  • Exercise 21:
  • Given \(H(-2,9)\) and \(M(8,0)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the other endpoint
  • \(x=2\times8-(-2)=16 + 2=18\)
  • ## Step2: Calculate the y - coordinate of the other endpoint
  • \(y=2\times0 - 9=-9\)
  • # Answer:
  • \(G(18,-9)\)
  • Exercise 22:
  • Given \(G(-4,1)\) and \(M(-\frac{13}{2},-6)\).
  • # Explanation:
  • ## Step1: Calculate the x - coordinate of the other endpoint
  • \(x=2\times(-\frac{13}{2})-(-4)=-13 + 4=-9\)
  • ## Step2: Calculate the y - coordinate of the other endpoint
  • \(y=2\times(-6)-1=-12 - 1=-13\)
  • # Answer:
  • \(H(-9,-13)\)