Question

1-3 lesson quiz. items 1-3 refer to the graph shown. 1. what is the midpoint of $overline{ab}$? 2. what is the length of $overline{ab}$? round your answer to the nearest hundredth. 3. what are the coordinates of the point $\frac{3}{4}$ of the way from a to b?

Explanation:

1. Mid - point formula:

• Let the coordinates of point \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). The mid - point \(M\) of the line segment \(\overline{AB}\) has coordinates \(M = (\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). First, we need to assume the coordinates of \(A\) and \(B\) from the graph. Let \(A=(3,5)\) and \(B=(7,1)\).
• Calculate the \(x\) - coordinate of the mid - point:
• \(x_m=\frac{3 + 7}{2}\)
• \(x_m=\frac{10}{2}=5\)
• Calculate the \(y\) - coordinate of the mid - point:
• \(y_m=\frac{5+1}{2}\)
• \(y_m=\frac{6}{2}=3\)
• The mid - point of \(\overline{AB}\) is \((5,3)\).

1. Distance formula:

• The distance \(d\) between two points \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\) is given by \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• Substitute \(x_1 = 3,y_1 = 5,x_2 = 7,y_2 = 1\) into the formula:
• \(d=\sqrt{(7 - 3)^2+(1 - 5)^2}\)
• First, calculate the values inside the square - root: \((7 - 3)^2=4^2 = 16\) and \((1 - 5)^2=(-4)^2 = 16\).
• Then \(d=\sqrt{16 + 16}=\sqrt{32}\approx5.66\)

1. Finding a point \(\frac{3}{4}\) of the way from \(A\) to \(B\):

• Let \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). The coordinates of a point \(P\) that is \(t\) of the way from \(A\) to \(B\) are given by \(P=(x_1+t(x_2 - x_1),y_1+t(y_2 - y_1))\), where \(t=\frac{3}{4}\).
• Calculate the \(x\) - coordinate of the point:
• \(x=x_1+t(x_2 - x_1)=3+\frac{3}{4}(7 - 3)\)
• \(x=3+\frac{3}{4}\times4=3 + 3=6\)
• Calculate the \(y\) - coordinate of the point:
• \(y=y_1+t(y_2 - y_1)=5+\frac{3}{4}(1 - 5)\)
• \(y=5+\frac{3}{4}\times(-4)=5-3 = 2\)
• The coordinates of the point \(\frac{3}{4}\) of the way from \(A\) to \(B\) are \((6,2)\)

Answer:

1. \((5,3)\)
2. Approximately \(5.66\)
3. \((6,2)\)