Question 127642
{{{drawing(500,500,-15,15,-15,15,
graph(500,500,-15,15,-15,15,0),
circle(-2,9,0.08),
circle(-2,9,0.1),
circle(-2,9,0.12),
circle(-4,-2,0.08),
circle(-4,-2,0.1),
circle(-4,-2,0.12),
circle(1,-12,0.08),
circle(1,-12,0.1),
circle(1,-12,0.12),
circle(3,-1,0.08),
circle(3,-1,0.1),
circle(3,-1,0.12),
line(-2,9,-4,-2),
line(-4,-2,1,-12),
line(1,-12,3,-1),
line(3,-1,-2,9))}}}


First let's find the slope of the line through the points (-2,9) and (1,-12) (these vertices are opposite from one other)





Let's denote the first point (-2,9) as *[Tex \Large \left(x_{1},y_{1}\right)]. In other words, *[Tex \LARGE x_{1}=-2] and *[Tex \LARGE y_{1}=9]


Now let's denote the second point (1,-12) as *[Tex \Large \left(x_{2},y_{2}\right)]. In other words, *[Tex \Large x_{2}=1] and *[Tex \Large y_{2}=-12]




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{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula


{{{m=(-12-9)/(1--2)}}} Plug in {{{y[2]=-12}}},{{{y[1]=9}}},{{{x[2]=1}}},{{{x[1]=-2}}}



{{{m=-21/3}}} Subtract the terms in the numerator {{{-12-9}}} to get {{{-21}}}.  Subtract the terms in the denominator {{{1--2}}} to get {{{3}}}

  

{{{m=-7}}} Reduce


  

So the slope of the line through the points (-2,9) and (1,-12) is {{{m=-7}}}





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Now let's find the slope of the line through the points (-4,-2) and (3,-1) (these vertices are opposite from one other)






Let's denote the first point (-4,-2) as *[Tex \Large \left(x_{1},y_{1}\right)]. In other words, *[Tex \LARGE x_{1}=-4] and *[Tex \LARGE y_{1}=-2]


Now let's denote the second point (3,-1) as *[Tex \Large \left(x_{2},y_{2}\right)]. In other words, *[Tex \Large x_{2}=3] and *[Tex \Large y_{2}=-1]




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{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula


{{{m=(-1--2)/(3--4)}}} Plug in {{{y[2]=-1}}},{{{y[1]=-2}}},{{{x[2]=3}}},{{{x[1]=-4}}}



{{{m=1/7}}} Subtract the terms in the numerator {{{-1--2}}} to get {{{1}}}.  Subtract the terms in the denominator {{{3--4}}} to get {{{7}}}

  



So the slope of the line through the points (-4,-2) and (3,-1) is {{{m=1/7}}}





Since the product of the two slopes is -1 (ie {{{-7(1/7)=-7/7=-1}}}), this shows us that the two slopes are perpendicular. So the two diagonals are also perpendicular. This shows us that the figure is a rhombus.