Question

task 4: given the points a(-3, 2), b(5, 2), c(1, 1), and d(1, 7), find the midpoint of ab, m_ab, and the midpoint of cd, m_cd. do the segments intersect at one of the midpoints? explain your reasoning. use the midpoint formula to calculate the coordinates of m_ab. m_ab=(\frac{x_a + x_b}{2},\frac{y_a + y_b}{2})=(\frac{-3 + 5}{2},\frac{2+2}{2})=(1, 2) use the midpoint formula to calculate the coordinates of m_cd. m_cd=(\frac{x_c + x_d}{2},\frac{y_c + y_d}{2})=(\frac{1 + 1}{2},\frac{1+7}{2})=(1, 4) a. why is the y - coordinate of the midpoint the same as the y - coordinate of both endpoints? b. why is the x - coordinate of the midpoint the same as the x - coordinate of both endpoints?

Explanation:

Step1: Recall mid - point formula

The mid - point 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: Calculate mid - point of $\overline{AB}$

Given $A(-3,2)$ and $B(5,2)$, using the mid - point formula $M_{\overline{AB}}=(\frac{-3 + 5}{2},\frac{2+2}{2})=(\frac{2}{2},\frac{4}{2})=(1,2)$.

Step3: Calculate mid - point of $\overline{CD}$

Given $C(1,1)$ and $D(1,7)$, using the mid - point formula $M_{\overline{CD}}=(\frac{1 + 1}{2},\frac{1+7}{2})=(\frac{2}{2},\frac{8}{2})=(1,4)$.

Step4: Analyze intersection at mid - points

The segments do not intersect at one of the mid - points. The $x$ - coordinates of the mid - points of $\overline{AB}$ and $\overline{CD}$ are the same ($x = 1$), but the $y$ - coordinates are different ($y = 2$ for $M_{\overline{AB}}$ and $y = 4$ for $M_{\overline{CD}}$). For two segments to intersect at a mid - point, both the $x$ and $y$ coordinates of the mid - points must be the same.

Step5: Answer part A

The $y$ - coordinate of the mid - point of $\overline{AB}$ is the same as the $y$ - coordinate of both endpoints because the $y$ - values of $A$ and $B$ are equal ($y_1=y_2 = 2$). When we use the formula $\frac{y_1 + y_2}{2}$, if $y_1=y_2$, then $\frac{y_1 + y_2}{2}=y_1=y_2$.

Step6: Answer part B

The $x$ - coordinate of the mid - point of $\overline{CD}$ is the same as the $x$ - coordinate of both endpoints because the $x$ - values of $C$ and $D$ are equal ($x_1=x_2 = 1$). When we use the formula $\frac{x_1 + x_2}{2}$, if $x_1=x_2$, then $\frac{x_1 + x_2}{2}=x_1=x_2$.

Answer:

$M_{\overline{AB}}=(1,2)$; $M_{\overline{CD}}=(1,4)$; The segments do not intersect at one of the mid - points.
A. Because the $y$ - values of $A$ and $B$ are equal.
B. Because the $x$ - values of $C$ and $D$ are equal.