Question

1. determine the coordinates of the midpoint of the segment with endpoints ( r(3,16) ) and ( s(7,-6) ).
2. determine the coordinates of the midpoint of the segment with endpoints ( w(-5,10.2) ) and ( x(12,4.5) ).
3. point ( c ) is the midpoint of ( overline{ab} ). point ( a ) has coordinates ( (2,4) ), and point ( c ) has coordinates ( (5,0) ).

a. what are the coordinates of point ( b )?
b. what is ( ab )?
c. what is ( bc )?

Explanation:

Step1: Midpoint formula

The midpoint formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).

Step2: Solve problem 15

For \(R(3,16)\) and \(S(7,-6)\), \(x=\frac{3 + 7}{2}=\frac{10}{2} = 5\), \(y=\frac{16+( - 6)}{2}=\frac{10}{2}=5\).

Step3: Solve problem 16

For \(W(-5,10.2)\) and \(X(12,4.5)\), \(x=\frac{-5 + 12}{2}=\frac{7}{2}=3.5\), \(y=\frac{10.2 + 4.5}{2}=\frac{14.7}{2}=7.35\).

Step4: Solve problem 17a

Let \(B=(x,y)\). Using midpoint formula: \(\frac{2 + x}{2}=5\) (for \(x\)-coordinate) and \(\frac{4 + y}{2}=0\) (for \(y\)-coordinate).
From \(\frac{2 + x}{2}=5\), we get \(2+x = 10\), \(x = 8\). From \(\frac{4 + y}{2}=0\), we get \(4+y=0\), \(y=-4\). So \(B=(8,-4)\).

Step5: Solve problem 17b

Distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(A(2,4)\) and \(B(8,-4)\), \(d=\sqrt{(8 - 2)^2+(-4 - 4)^2}=\sqrt{36 + 64}=\sqrt{100}=10\).

Step6: Solve problem 17c

Since \(C\) is the mid - point, \(BC=\frac{AB}{2}\). So \(BC = 5\).

Answer:

1. \((5,5)\)
2. \((3.5,7.35)\)

17a. \((8,-4)\)
17b. \(10\)
17c. \(5\)