The points A(-1, -2), B(4,3), C(2,5) and D(-3,0) form a ___
Rectangle
We know that, distance between the points (x1,y1) and (y2,y2)
d=√(x2−x1)2+(x2−y1)2
So, distance between A(-1,-2) and B(4, 3)
AB=√(4+1)2+(3+2)2=√52+52=√25+25=5√2
Distance between C(2,5) and D(-3,0)
CD=√(−3−2)2+(0−5)2=√(−5)2+(−5)2=√25+25=5√2
Distance between A(-1,-2) and D(-3,0),
AD=√(−3+1)2+(0+2)2=√(−2)+22=√4+4=2√2
Similarly, BC = 2√2
We know that, in a rectangle, opposite sides are equal and diagonals are equal and they bisect each other.
Since, AB = CD and AD = BC
Distance between A(-1,-2) and C(2,5)
AC=√(2+1)2+(5+2)2=√32+72=√9+49=√58
And distance between D(-3,0) and B(4,3),
DB=√(4+3)2+(3−0)2=√72+32=√49+9=√58
Since, the diagonals AC and BD are equal.
the points A(-1,-2), B(4,3), C(2,5) and D(-3,0) form a rectangle.