Properties of Quadrilaterals

A Quadrilaterals are identified as 4-sided polygons, whose interior angles always add up to \(360°\).
There are 5 distinct types of quadrilaterals, each with their own unique sets of features: Rectangles, Squares, Parallelograms, Rhombuses, and Trapezoids.

Square

• Contains 4 equal sides, all of which are parallel to each other
• Has 4 right angles, meaning that each angle is \(90°\)
• Diagonals bisect each other perpendicularly

Rectangle

• Contains 2 pairs of adjacent sides, each of which are equal and parallel to each other
• Has 4 right angles, meaning that each angle is \(90°\)
• Diagonals bisect each other

Parallelogram

• Are formed within the midsegments of any quadrilateral
• Contains 2 pairs of adjacent sides both of which are parallel and equal to each other
• Opposite angles are equal to each other
• Diagonals bisect each other

Rhombus

• All 4 sides are equal
• Contains 2 pairs of adjacent sides both of which are parallel and equal to each other
• Diagonals bisect each other perpendicularly

Trapezoid

• Has only one pair of parallel sides, referred to as bases
• Contains a pair of lateral sides, which are positioned at an angle and not parallel to each other
• The midsegment is parallel to the parallel sides; its length is the average of those sides

Show Answer

1. We can use the midpoint formula to find the midpoints of each of the respective side lengths:

\(\text{midpoint}^{(\text{D})} = (\cfrac{-2 - 4)}{2}, \cfrac{4 - 2}{2}) = (\cfrac{-6}{2}, \cfrac{2}{2}) = (-3, 1)\)

\(\text{midpoint}^{(\text{E})} = (\cfrac{-4 + 2)}{2}, \cfrac{-2 - 4}{2}) = (\cfrac{-2}{2}, \cfrac{-6}{2}) = (-1, -3)\)

\(\text{midpoint}^{(\text{F})} = (\cfrac{2 + 4)}{2}, \cfrac{-4 + 0}{2}) = (\cfrac{6}{2}, \cfrac{-4}{2}) = (3, -2)\)

\(\text{midpoint}^{(\text{G})} = (\cfrac{-2 + 4)}{2}, \cfrac{4 + 0}{2}) = (\cfrac{2}{2}, \cfrac{4}{2}) = (1, 2)\)

Therefore, we can determine that the respecive midpoints are \(\text{D}(-3,1)\), \(\text{E}(-1,-3)\), \(\text{F}(3,-2)\) and \(\text{G}(1,2)\).

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2. To find that the opposite sides are parallel, we can find the respective slopes for each side length. In this instance, \(\text{DE}\) is opposite to \(\text{FG}\) and \(\text{EF}\) is opposite to \(\text{DG}\):

\(\text{slope}^{(\text{DE})} = \cfrac{-3 - 1}{-1 - (-3)} = \cfrac{-4}{2} = -2\)

\(\text{slope}^{(\text{EF})} = \cfrac{-2 - (-3)}{3 - (-1)} = \cfrac{1}{4}\)

\(\text{slope}^{(\text{FG})} = \cfrac{2 - (-2)}{1 - 3} = \cfrac{4}{-2} = -2\)

\(\text{slope}^{(\text{DG})} = \cfrac{1 - 2}{-3 - 1} = \cfrac{-1}{-4} = \cfrac{1}{4}\)

As the slopes of \(\text{DE} = \text{FG}\) and \(\text{EF} = \text{DG}\), we can determine that the opposite sides are parallel to each other.

We can verify that the opposite sides are equal in length by using the length of a line formula:

\(\text{Length}^{(\text{DE})} = \sqrt{(-1 - (-3))^2 + (-3 - 1)^2} = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20}\)

\(\text{Length}^{(\text{EF})} = \sqrt{(3 - (-1))^2 + (-2 - (-3))^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{17}\)

\(\text{Length}^{(\text{FG})} = \sqrt{(1 - 3)^2 + (2 - (-2))^2} = \sqrt{(-2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20}\)

\(\text{Length}^{(\text{DG})} = \sqrt{(1 - (-3))^2 + (2 - 1)^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17}\)

As the side lengths of \(\text{DE} = \text{FG}\) and \(\text{EF} = \text{DG}\), we can determine that the opposite sides are equal in length.

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3. We can determine that \(\text{DEFG}\) is a parallelogram since none of the slopes are negative reciprocols.

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4. Based on the coordinates we have determined from the previous questions, we can graph both \(\text{STUV}\) and \(\text{DEFG}\):