Question

for items 7 - 8, suppose \\( \overline { x y } \\) has one endpoint at \\( x ( 0,0 ) \\).

1. if \\( ( 3,4 ) \\) is the midpoint of \\( \overline { x y } \\), what are the coordinates of point \\( y \\)?

(, )

1. what are the coordinates of \\( y \\) if \\( ( 3,4 ) \\) is \\( \frac { 1 } { 3 } \\) of the way from \\( x \\) to \\( y \\)?

(, )

Explanation:

Step1: Recall the midpoint formula

The midpoint formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Let \(X(0,0)\) be \((x_1,y_1)\) and \(Y(x,y)\) be \((x_2,y_2)\), and midpoint \((3,4)\).

Step2: Solve for \(x\)

Using the \(x -\)coordinate of midpoint formula: \(\frac{0 + x}{2}=3\), then \(x = 6\).

Step3: Solve for \(y\)

Using the \(y -\)coordinate of midpoint formula: \(\frac{0 + y}{2}=4\), then \(y = 8\).

Step4: Recall the section formula

If a point \(P(x,y)\) divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here \(X(0,0)\) is \((x_1,y_1)\), \(Y(x,y)\) is \((x_2,y_2)\), \(m = 1\), \(n=2\) (since the point \((3,4)\) is \(\frac{1}{3}\) of the way from \(X\) to \(Y\), so the ratio of \(XP:PY=1:2\)), and \(P(3,4)\).

Step5: Solve for \(x\) in section formula

\(3=\frac{1\times x+2\times0}{1 + 2}\), then \(9=x\).

Step6: Solve for \(y\) in section formula

\(4=\frac{1\times y+2\times0}{1 + 2}\), then \(12=y\).

Answer:

1. \((6,8)\)
2. \((9,12)\)