Question

# Question 12 The points A(-1, -2), B(4,3), C(2,5) and D(-3,0) in that order form a rectangle.

Solution

## True Distance between A(-1,-2) and B(4, 3) AB=√(4+1)2+(3+2)2=√52+52=√25+25=5√2Distance 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 the points(x1,y1) and (y2,y2)d=√(x2−x1)2+(x2−y1)2]Distance between A(-1,-2) and D(-3,0),AD=√(−3+1)2+(0+2)2=√(−2)+22=√4+4=2√2 Distance between B(4,3) and C(2,5),BC=√(2−4)2+(5−3)2=√(−2)+22=√4+4=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 Also, 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, diagonals AC and BD are equal. Hence, the points A(-1,-2), B(4,3), C(2,5) and D(-3,0) form a rectangle.

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