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Complete Question
Given quadrilateral EFGH with vertices at E(-4,8), F(8,4), G(5,-5) and H(-7,-1), prove using coordinate geometry that EFGH is a rectangle.
Answer:
Quadrilateral EFGH is a rectangle.l because:
EF = GH and FG = EH
Explanation:
The formula for coordinate geometry is given as :
√(x2 - x1)² + (y2 - y1)² when we have coordinates: (x1, y1) and (x2 , y2)
For the quadrilateral EFGH with given coordinates above to be a rectangle,
EF = GH
FG = EH
Hence:
For side EF
E(-4,8), F(8,4)
= √(8 - (-4))² + (4 - 8)²
= √12² + -4²
= √144 + 16
= √160 units
For side FG
F(8,4), G(5,-5)
=√(5 - 8)² + (-5 - 4)²
= √-3² + -9²
= √9 + 81
= √90 units
For Side GH
G(5,-5) , H(-7,-1)
= √(-7 - 5)² + (-1 - (-5))²
= √-12² + 4²
= √144 + 16
= √160 units
For side EH
E(-4,8), H(-7,-1)
= √(-7 -(-4))² +(-1 - 8)²
= √-3² + -9²
= √9 + 81
= √90 units
From the above calculation, we can see that truly,
EF = GH
FG = EH
Therefore, quadrilateral EFGH is a rectangle.