Is the Quadrilateral a Square? Yes. Explanation: See solution.
Practice makes perfect
Let's begin by writing the coordinates of the vertices of the given quadrilateral.
We are told that a square is a quadrilateral that has opposite sides that are parallel, consecutive sides that are perpendicular, and diagonals that are perpendicular. Let's see if these things happen in the given quadrilateral.
Opposite and Consecutive Sides
In the diagram, we can see that vertices E and F have the same y-coordinate. Therefore, EF is a horizontal segment. Because of the same reason, GH is also a horizontal segment. Similarly, vertices F and G have the same x-coordinate. Therefore, FG is a vertical segment, and HE is also a vertical segment.
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Horizontal & & Vertical
Segments & & Segments [0.8em]
EF and GH & & FG and HE
All vertical segments are parallel, and all horizontal segments are parallel. This means that the opposite sides of the given quadrilateral are parallel. Furthermore, vertical and horizontal segments are perpendicular to each other. Therefore, the consecutive sides are perpendicular.
Diagonals
Finally, we need to determine if the diagonals are perpendicular. To do so, we will check if the product of their slopes equals - 1. Let's first calculate the slope of each diagonal starting with HF. To do so, we will substitute (3,3) and (- 4,- 4) into the Slope Formula.
We found that the slope of HF is 1 and that the slope of EG is - 1. Let's calculate their product.
m_(HF)* m_(EG)= 1(- 1)=- 1
The diagonals are perpendicular. This, and the fact that the opposite sides are parallel and the consecutive sides are perpendicular, leads us to the conclusion that the quadrilateral shown is a square.
