Question

what is the mid - point for a segment with the following endpoints? (4,5) and (10, - 3) ((x_1 + x_2)/2,(y_1 + y_2)/2) midpoint ________ 8. what is the mid - point for a segment with the following endpoints? (-2,-3) and (-11,20) ((x_1 + x_2)/2,(y_1 + y_2)/2) midpoint ________ 9. calculate the length between the following points using the distance formula (1,6) and (7,14) sqrt((x_2 - x_1)^2+(y_2 - y_1)^2) length: ________ 10. calculate the length between the following points using the distance formula (-3,-2) and (9,3) sqrt((x_2 - x_1)^2+(y_2 - y_1)^2) length: ________ optional: complete problems 10, 11, 13, 15, 17 - 20, 24, 29 from your textbook on separate lined p when you are finished, check the odd problems for solutions at the back of the textbook.

Explanation:

Step1: Identify mid - point formula and values for (4,5) and (10, - 3)

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Here \(x_1 = 4\), \(x_2=10\), \(y_1 = 5\), \(y_2=-3\).

Step2: Calculate x - coordinate of mid - point

\(\frac{x_1 + x_2}{2}=\frac{4 + 10}{2}=\frac{14}{2}=7\)

Step3: Calculate y - coordinate of mid - point

\(\frac{y_1 + y_2}{2}=\frac{5+( - 3)}{2}=\frac{5 - 3}{2}=1\)

Step4: Identify mid - point formula and values for (-2,-3) and (-11,20)

For points \((x_1,y_1)=(-2,-3)\) and \((x_2,y_2)=(-11,20)\), use the mid - point formula \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\)

Step5: Calculate x - coordinate of mid - point

\(\frac{x_1 + x_2}{2}=\frac{-2+( - 11)}{2}=\frac{-2-11}{2}=-\frac{13}{2}=-6.5\)

Step6: Calculate y - coordinate of mid - point

\(\frac{y_1 + y_2}{2}=\frac{-3 + 20}{2}=\frac{17}{2}=8.5\)

Step7: Identify distance formula and values for (1,6) and (7,14)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 1\), \(x_2=7\), \(y_1 = 6\), \(y_2=14\)

Step8: Calculate the distance

\[

$$\begin{align*} d&=\sqrt{(7 - 1)^2+(14 - 6)^2}\\ &=\sqrt{6^2+8^2}\\ &=\sqrt{36 + 64}\\ &=\sqrt{100}\\ &=10 \end{align*}$$

\]

Step9: Identify distance formula and values for (-3,-2) and (9,3)

For points \((x_1,y_1)=(-3,-2)\) and \((x_2,y_2)=(9,3)\), use \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Step10: Calculate the distance

\[

$$\begin{align*} d&=\sqrt{(9-( - 3))^2+(3-( - 2))^2}\\ &=\sqrt{(9 + 3)^2+(3 + 2)^2}\\ &=\sqrt{12^2+5^2}\\ &=\sqrt{144+25}\\ &=\sqrt{169}\\ &=13 \end{align*}$$

\]

Answer:

• Mid - point for \((4,5)\) and \((10,-3)\): \((7,1)\)
• Mid - point for \((-2,-3)\) and \((-11,20)\): \((-6.5,8.5)\)
• Length between \((1,6)\) and \((7,14)\): \(10\)
• Length between \((-3,-2)\) and \((9,3)\): \(13\)