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17 days ago

Find the midpoint of the segment with endpoints of −11 + i and −4 + 4i.

Mathematics

2 answers:

Leona [9.2K]17 days ago

7 0

(-11 + -4)/2 = -15/2

(i + 4i)/2 = 5/2 i

the midpoint calculates to (-15/2, 5/2 i)

or in decimal form (-7.5, 2.5i)

tester [8.8K]17 days ago

5 0

$\bf \begin{cases} -11+i\implies -11+1i\\ \qquad (-11,1)\\ -4+4i\\ \qquad (-4,4) \end{cases} \\\\\\ ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ (\stackrel{x_1}{-11}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{-4}~,~\stackrel{y_2}{4}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~,~~~ \cfrac{ y_2 + y_1}{2} \right)$

$\bf \left( \cfrac{-4-11}{2}~~,~~ \cfrac{4+1}{2}\right)\implies \left(-\frac{15}{2}~,~\frac{5}{2} \right)\implies \left(-7\frac{1}{2}~,~2\frac{1}{2} \right) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill -7\frac{1}{2}+2\frac{1}{2}i~\hfill$

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It's known that

When two lines are parallel, their slopes are identical.

The slope between any two points can be calculated using the following formula:

$m=\frac{y2-y1}{x2-x1}$

We will calculate the slope for each case to find the solution to the problem.

Case A) Point $B(5,2)\ C(7,-5)$

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Insert the values into the formula:

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$m=\frac{-7}{2}$

$m=-3.5$

Thus,

The equation $y=-3.5x-15$ is parallel to the line that goes through the points $B(5,2)\ C(7,-5)$

Therefore,

the result for Part A) is

$B(5,2)\ C(7,-5)$ ------> $y=-3.5x-15$

Case B) Point $D(11,6)\ E(5,9)$

Calculate the slope of DE

Plug the values into the formula:

$m=\frac{9-6}{5-11}$

$m=\frac{3}{-6}$

$m=-0.5$

Thus,

The equation $y=-0.5x-3$ is parallel to the line that goes through the points $D(11,6)\ E(5,9)$

Therefore,

the result for Part B) is

$D(11,6)\ E(5,9)$ ------> $y=-0.5x-3$

Case C) Point $F(-7,12)\ G(3,-8)$

Determine the slope of FG

Insert the values into the formula:

$m=\frac{-8-12}{3+7}$

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$m=-2$

Thus,

Any linear equation with slope $m=-2$ will be parallel to the line through the points $F(-7,12)\ G(3,-8)$

Case D) Point $H(4,4)\ I(8,9)$

Calculate the slope of HI

Substitute the values in the formula:

$m=\frac{9-4}{8-4}$

$m=\frac{5}{4}$

$m=1.25$

Thus,

The equation $y=1.25x+4$ is parallel to the line through the points $H(4,4)\ I(8,9)$

Therefore,

the result for Part D) is

$H(4,4)\ I(8,9)$ ------> $y=1.25x+4$

Case E) Point $J(7,2)\ K(-9,8)$

Determine the slope of JK

Insert the values into the formula:

$m=\frac{8-2}{-9-7}$

$m=\frac{6}{-16}$

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Thus,

Any linear equation characterized by slope $m=-0.375$ will be parallel to the line that runs through the points $J(7,2)\ K(-9,8)$

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Find the slope of LM

Substitute the values in the formula:

$m=\frac{-12+7}{4-5}$

$m=\frac{-5}{-1}$

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Thus,

The equation $y=5x+19$ is parallel to the line connecting the points $L(5,-7)\ M(4,-12)$

Therefore,

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Information provided

$X_{1}=688$ denote the number of men possessing smartphones

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$n_{1}=989$ group of men sampled

$n_{2}=1012$ group of women sampled

$p_{1}=\frac{688}{989}=0.696$ symbolize the proportion of men with smartphones

$p_{2}=\frac{671}{1012}=0.663$ symbolize the proportion of women with smartphones

$\hat p$ denote the pooled estimate of p

z would denote the test statistic

$p_v$ signify the value

Part a

The objective is to evaluate if there is a disparity in smartphone ownership between men and women; the hypothesis statements would be:

Null hypothesis: $p_{1} = p_{2}$

Alternative hypothesis: $p_{1} \neq p_{2}$

Part b

The statistic relevant to this case is expressed as:

$z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}$ (1)

Where $\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{688+671}{989+1012}=0.679$

Part c

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Part d

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