The side of square ABCD is 6. The midpoints of AB and BC are E and F
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16 May 2021, 08:49
(Step 1) since E and F are the midpoints on their respective sides of the square ——-> EB = (1/2) (6) = BF
Right triangle EBF is thus a right isosceles triangle with Legs EB and BF = 3 and Hypotenuse EF = 3*sqrt(2)
(Step 2) we are told that Point G is the Midpoint on the Hypotenuse EF so:
EG = (1.5) sqrt(2) = GF
(Step 3) drawing the Diagonal of the entire Square connecting Vertex B to Vertex D, we find that Diagonal BD = (6) * sqrt(2)
Further more, GB (which lies on Diagonal BD and travels to Vertex B of the right isosceles triangle) is the Median between the 2 equal sides of the right isosceles triangle: it bisects the hypotenuse
Rule: the Median drawn from the vertex between the 2 equal sides of an isosceles triangle to the non-equal side is a Line of Symmetry of the isosceles triangle. Thus:
Median BG = Perpendicular Bisector = Altitude = Angle Bisector
This Median BG subdivides right triangle EBF into another two 45-45-90 right isosceles triangles.
Using the Ratio of Side Lengths for a 45-45-90 right triangle,
the length of GB = (1.5) sqrt(2) = EG
(Step 4) Entire Square Diagonal DB is composed of the Diagonal of the Square HGID and Median GB in the right isosceles triangle.
DB = DG + GB
We know DB = (6) sqrt(2) ——- since it is the diagonal for the entire square of side length 6
And we know GB = (1.5) sqrt(2) ——- as found above
Thus:
DG = DB - GB
DG = (6) sqrt(2) - (1.5) sqrt(2)
DG = (4.5) sqrt(2)
DG = the diagonal of square HGID
We are looking for the side length of this square HGID, namely side DI
Rule: the Diagonal of a square = (side) * sqrt(2)
Since the Diagonal DG = (4.5) * sqrt(2)
The Side DI = 4.5
Answer:
4.5
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