Answer:
The quadrilateral ABCD is a square.
Step-by-step explanation:
According to the statement, AB, BC, CD and DA are the sides of the quadrilateral, whereas DB and AC are its diagonals. If we know that [tex]x = 7[/tex], [tex]AB = 4\cdot x - 2[/tex], [tex]BC = 2\cdot x + 12[/tex], [tex]DC = 3\cdot x +5[/tex], [tex]AD = 6\cdot x -16[/tex], [tex]AC = 4\cdot x + 12[/tex] and [tex]DB = 8\cdot x -10[/tex], the lengths of each line segment are respectively:
Sides
[tex]AB = 4\cdot (7) -2[/tex]
[tex]AB = 26[/tex]
[tex]BC = 2\cdot (7) +12[/tex]
[tex]BC = 26[/tex]
[tex]DC = 3\cdot (7) +5[/tex]
[tex]DC = 26[/tex]
[tex]AD = 6\cdot (7) -16[/tex]
[tex]AD = 26[/tex]
Diagonals
[tex]AC = 4\cdot (7) +12[/tex]
[tex]AC = 40[/tex]
[tex]DB = 8\cdot (7) - 10[/tex]
[tex]DB = 40[/tex]
This information indicates that this quadrilateral is a square because of these characteristics:
1) All sides have the same length.
2) The ratio of any diagonal to any side is [tex]\sqrt{2}[/tex].
