Quadrilateral EFGH has vertices at E(10, 4), F(16, 4), G(16, 10) and H(10, 10). Based on the properties of the diagonals, is quadrilateral EFGH a rectangle, rhombus, or square? Use the distance and slope formulas to prove your conclusion. Show your work.
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Answers
Given : Quadrilateral EFGH has vertices at E(10, 4), F(16, 4), G(16, 10) and H(10, 10).
To Find : Based on the properties of the diagonals, is quadrilateral EFGH a rectangle, rhombus, or square?
Use the distance and slope formulas to prove your conclusion.
Solution:
E(10, 4), F(16, 4), G(16, 10) and H(10, 10).
Diagonal
EG & HF
Mid point of EG = ( 10 + 16)/2 , ( 4 + 10)/2
= 13 , 7
Mid point of HF = ( 16 + 10)/2 , ( 4 + 10)/2
= 13 , 7
Mid point is common
Hence both diagonal bisect each other
EG = √(16 - 10)² + (10 - 4)² = 6√2
HF = √(16 - 10)² + (10 - 4)² = 6√2
EG = HF
Hence its a rectangle
Slope of EG = (10 - 4)/(16 - 10) = 6/6 = 1
Slope of HF = (10 - 4)/(10 - 16) = 6/-6 = -1
multiplication of slopes = - 1
Hence diagonals are perpendicular
Hence its a rhombus and square
quadrilateral EFGH is a SQUARE
which is always rectangle , rhombus and parallelogram also
EF = 6
FG = 6
GH = 6
EH = 6
Slope of EF = 0
Slope of FG = 6/0
Slope of GH = 0
Slope of EH = 6/0
Hence Verified that its a square
as all sides are Equal and adjacent sides are perpendicular
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