Answer:
A (2, 0) , B (3, 2) , C (6, 3) , D (5, 1)
The vertices of a parallelogram with non-perpendicular adjacent sides
A (3, 3) , B (3, 6) , C (7, 6) , D (7, 3)
The vertices of a rectangle with non-congruent adjacent sides
A (2, -2) , B (3, 0) , C (4, -2) , D (3, -4)
The vertices of a rhombus with non-perpendicular adjacent sides
A (3, 3) , B (2, 5) , C (4, 6) , D (5, 4)
The vertices of a square
Explanation:
* To find the vertices of each quadrilateral we will use the rule
of the slope and the rule of the distance to find the parallel
sides and perpendicular sides and the equal sides in length
- Slope = (y2 - y1)/(x2 - x1) = vertical change/horizontal change
- Distance = √[(x2 - x1)² + (y2 - y1)²]
* A (2, 0) , B (3, 2) , C (6, 3) , D (5, 1)
- Slope of AB = (2 - 0)/(3 - 2) = 2/1 = 2
- Slope of BC = (3 - 2)/(6 - 3) = 1/3
- Slope of CD = (1 - 3)/(5 - 6) = -2/-1 = 2
∵ Parallel lines have same slope
∴ AB // CD and BC // AD
∵ In any quadrilateral if every two opposite sides are parallel
then the quadrilateral is a parallelogram
∴ ABCD is a parallelogram
∵ AB and BC are two adjacent sides
∵ The product of the slopes of perpendicular lines = -1
∵ Slope AB × Slope BC = 2 × 1/3 = 2/3 ≠ -1
∴ AB and BC are not perpendicular
∴ ABCD is a parallelogram with non-perpendicular adjacent sides
* A (3, 3) , B (3, 6) , C (7, 6) , D (7, 3)
∵ A and B have the same x-coordinates
∴ AB is a vertical side
∵ B and C have the same y-coordinates
∴ BC is a horizontal side
∴ AB ⊥ BC ⇒ (1)
* Similar C and D have the same x-coordinates
∴ CD is a vertical side
∵ D and A have the same y-coordinate
∴ DA is a horizontal side
∴ CD ⊥ DA ⇒ (2)
* From (1) and (2)
∴ AB // CD and BC // AD
∴ AB ⊥ BC and CD ⊥ DA
∴ In quadrilateral ABCD every two opposite sides are parallel
and every two adjacent sides are perpendicular
∵ The length of AB = 6 - 3 = 3 units ⇒ vertical line
∵ The length of BC = 7 - 3 = 4 units ⇒ horizontal line
∴ AB ≠ BC in length ⇒ two adjacent sides
∴ ABCD is a rectangle with non-congruent adjacent sides
* A (2, -2) , B (3, 0) , C (4, -2) , D (3, -4)
- Slope AB = (0 - -2)/(3 - 2) = 2/1 = 2
- Slope BC = (-2 - 0)/(4 - 3) = -2/1 = -2
- Slope CD = (-4 - -2)/(3 - 4) = -2/-1 = 2
- Slope of DA = (-4 - -2)/(3 - 2) = -2/1 = -2
∴ AB // CD and BC // AD ⇒ (1)
∵ AB = √[(3 - 2)² + (0 - -2)²] = √1 + 4 = √5
∵ BC = √[(4 - 3)² + (-2 - 0)²] = √1 + 4 = √5
∵ CD = √[(3 - 4)² + (-4 - -2)²] = √1 + 4 = √5
∵ AD = √[(3 - 2)² + (-4 - -2)²] = √1 + 4 = √5
∴ AB = BC = CD = DA ⇒ (2)
* From (1) and (2)
∵ Every two opposite sides are parallel
∵ All sides are equal in length
∴ ABCD is a rhombus
∵ Slope AB × slope BC = 2 × -2 = -4 ≠ 1
∴ AB and BC are not perpendicular
∴ ABCD is a rhombus with non-perpendicular adjacent sides
* A (3, 3) , B (2, 5) , C (4, 6) , D (5, 4)
- Slope of AB = (5 - 3)/(2 - 3) = 2/-1 = -2
- Slope of BC = (6 - 5)/(4 - 2) = 1/2
- Slope of CD = (4 - 6)/(5 - 4) = -2/1 = -2
- Slope of AD = (4 - 3)/(5 - 2) = 1/2
∴ AB // CD and BC // AD
∵ Slope AB × slope BC = -2 × 1/2 = -1
∴ AB ⊥ BC
∵ Slope CD × slope AD = -2 × 1/2 = -1
∴ CD ⊥ AD
∵ AB = √[(2 - 3)² + (5 - 3)²] = √1 + 4 = √5
∵ BC = √[(4 - 2)² + (6 - 5)²] = √4 + 1 = √5
∵ CD = √[(5 - 4)² + (4 - 6)²] = √1 + 4 = √5
∵ AD = √[(5 - 3)² + (4 - 3)²] = √4 + 1 = √5
∴ AB = BC = CD = DA
* In quadrilateral ABCD:
∵ Every two opposite sides are parallel
∵ Every two adjacent sides are perpendicular
∵ All sides are equal in length
∴ ABCD is a square