Question 1203123
A quadrilateral has vertices at 

P ({{{-8}}}, {{{15}}}), Q ({{{10}}}, {{{11}}}), R ({{{12}}},{{{ -1}}}), and S ({{{-4}}}, {{{-9}}})



a.) What shape is this quadrilateral?



{{{ drawing( 600, 600, -20, 20, -20, 20,

 circle(-8,15,.15),locate(-8,16,P),
circle(10,11,.15),locate(10,12,Q),
circle(12,-1,.15),locate(12,-2,R),
circle(-4,-9,.15),locate(-4,-10,S),

blue(line(-8,15,-4,-9)),

blue(line(10,11,12,-1)),
blue(line(-8,15,10,11)),

blue(line(12,-1,-4,-9)),

graph( 600, 600, -20, 20, -20, 20, 0)) }}}

it could be a trapezoid or a trapezium

 a trapezoid when it has equal angles from a parallel side
a trapezium  is a quadrilateral with NO parallel sides



check if {{{ PS }}}and {{{ QR }}}are a pair of {{{ parallel}}} sides

Find the slope of the line through the points {{{ P}}} and {{{ S}}}, and {{{ Q }}}and {{{ R}}} as well.

if slopes are equal, than {{{ PS}}} and {{{ QR}}} are a pair of parallel sides

{{{ PS }}}slope is: {{{ m=(y2-y1)/(x2-x1)=(-9-15)/(-4+8)=-24/4=-6}}}

and 

{{{ QR}}} slope is: {{{ m=(y2-y1)/(x2-x1)=(-1-11)/(12-10)=-12/2=-6}}}


slopes are equal => {{{PS}}} and {{{QR }}}are a pair of parallel sides

so, we have a {{{trapezoid }}}


b.) Do the diagonals bisect each other?

a trapezoid can have only one pair of parallel sides, means that the diagonals do NOT bisect each other 



c.) What shape is formed by connecting the midsegments? (midsegment is a line that connects midpoints)


first find midpoints of each side

P ({{{-8}}}, {{{15}}}), Q ({{{10}}}, {{{11}}}), R ({{{12}}},{{{ -1}}}), and S ({{{-4}}}, {{{-9}}})

PQ: M1 ({{{(-8+10)/2}}}, {{{(15+11)/2}}})=({{{1}}}, {{{13}}})

QR :M2 ({{{(10+12)/2}}}, {{{(11-1)/2}}})=({{{11}}}, {{{5}}})

RS :M3 ({{{(12-4)/2}}}, {{{(-1-9)/2}}})=({{{4}}}, {{{-5}}})

PS:M4  ({{{(-8-4)/2}}}, {{{(15-9)/2}}})=({{{-6}}}, {{{3}}})



{{{ drawing( 600, 600, -20, 20, -20, 20,

 circle(-8,15,.15),locate(-8,16,P),
circle(10,11,.15),locate(10,12,Q),
circle(12,-1,.15),locate(12,-2,R),
circle(-4,-9,.15),locate(-4,-10,S),

blue(line(-8,15,-4,-9)),

blue(line(10,11,12,-1)),
blue(line(-8,15,10,11)),

blue(line(12,-1,-4,-9)),

circle(1,13,.25),locate(1,13,M1),
circle(11,5,.25),locate(11,5,M2),
circle(4,-5,.25),locate(4,-5,M3),
circle(-6,3,.25),locate(-6,3,M4),
green(line(1,13,11,5)),
green(line(1,13,-6,3)),
green(line(11,5,4,-5)),
green(line(-6,3,4,-5)),

graph( 600, 600, -20, 20, -20, 20, 0)) }}}


leaving for you to answer what shape is formed by connecting the midsegments