Question

What type of quadrilateral is quadrilateral ABCD if AB=4x-2, BC=2x+12, DC=3x+5, AD=6x-16, AC=4x+12, DB=8x-10, and x=7? Explain and use the properties of the types of quadrilaterals as proofs.

Answer 1

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 $x = 7$, $AB = 4\cdot x - 2$, $BC = 2\cdot x + 12$, $DC = 3\cdot x +5$, $AD = 6\cdot x -16$, $AC = 4\cdot x + 12$ and $DB = 8\cdot x -10$, the lengths of each line segment are respectively:

Sides

$AB = 4\cdot (7) -2$

$AB = 26$

$BC = 2\cdot (7) +12$

$BC = 26$

$DC = 3\cdot (7) +5$

$DC = 26$

$AD = 6\cdot (7) -16$

$AD = 26$

Diagonals

$AC = 4\cdot (7) +12$

$AC = 40$

$DB = 8\cdot (7) - 10$

$DB = 40$

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 $\sqrt{2}$.