Answer:
Its diagonals are perpendicular, then it is a rhombus
Step-by-step explanation:
* Lets revise the properties of the parallelogram and the rhombus
- In the parallelogram each two opposite sides are parallel
- In the parallelogram each two opposite sides are equal in length
- In the parallelogram the diagonals bisect each other
- In the rhombus each two opposite sides are parallel
- In the rhombus all the sides are equal in length
- In The rhombus the diagonals are perpendicular to each other
- Parallelogram is a rhombus if two adjacent sides are equal in length
- Parallelogram is a rhombus if its two diagonals are perpendicular
* Now lets solve the problem
∵ The vertices of the parallelogram are
A(-3 , 2) , B(-2 , 6) , C (2 , 7) , D (1 , 3)
- The slope of the line which passes through points (x1 , y1) and (x2 , y2)
is m = (y2 - y1)/(x2 - x1)
* lets find the slopes of the sides and the diagonals
∵ The slope of AB = (6 - 2)/(-2 - -3) = 4/1 = 4
∵ The slope of BC = (7 - 6)/(2 - -2) = 1/4 = 1/4
∵ The slope of CD = (3 - 7)/(1 - 2) = -4/-1 = 4
∵ The slope of DA = (2 - 3)/(-3 - 1) = -1/-4 = 1/4
- The product of the slopes of the perpendicular line is -1
∵ AB and BC are two adjacent sides and the product of their slopes
= 4 × 1/4 = 1 ≠ -1
∴ AB and BC are not perpendicular
∴ The parallelogram can not be a rectangle
- Lets check the slopes of the diagonals
∵ The diagonals of the parallelogram are AC and BD
∵ The slope of AC = (7 - 2)/(2 - -3) = 5/5 = 1
∵ The slope of BD = (3 - 6)/(1 - -2) = -3/3 = -1
∵ The product of the slopes of AC and BD = 1 × -1 = -1
∴ AC and BD are perpendicular
∴ ABCD is a rhombus
