show that (-1,1), (9,4), (20,6) and (10,1) are the vertices of a rhombus and then find the area of this rhombus?

no they are not

just sketch it and find the slopes of the lines, none are equal

In a rhombus opposite sides should have the same slope and they must be equal in length.

To show that the given points (-1, 1), (9, 4), (20, 6), and (10, 1) are the vertices of a rhombus, we need to demonstrate two things:

1) All four sides of the quadrilateral formed by these points are congruent (equal in length).
2) The diagonals of the quadrilateral are perpendicular to each other.

Step 1: Find the lengths of all four sides.
Using the distance formula, the distance between two points (x1, y1) and (x2, y2) is given by:

distance = √[(x2 - x1)^2 + (y2 - y1)^2]

Using this formula, we can calculate the length of each side:

Side AB:
distance AB = √[(9 - (-1))^2 + (4 - 1)^2] = √[10^2 + 3^2] = √109

Side BC:
distance BC = √[(20 - 9)^2 + (6 - 4)^2] = √[11^2 + 2^2] = √125

Side CD:
distance CD = √[(10 - 20)^2 + (1 - 6)^2] = √[(-10)^2 + (-5)^2] = √125

Side DA:
distance DA = √[(-1 - 10)^2 + (1 - 1)^2] = √[(-11)^2 + 0^2] = √121 = 11

Step 2: Check if all sides are congruent.
We can see that the lengths of all four sides are equal:
AB = BC = CD = DA = √125.

Step 3: Find the slope of each diagonal.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:

slope = (y2 - y1) / (x2 - x1)

Using this formula, we can calculate the slopes of the two diagonals:

Diagonal AC:
slope AC = (6 - 1) / (20 - (-1)) = 5 / 21

Diagonal BD:
slope BD = (4 - 1) / (9 - 10) = 3 / (-1) = -3

Step 4: Check if the diagonals are perpendicular.
Two lines are perpendicular if and only if the product of their slopes is -1.

Let's check:

slope AC * slope BD = (5/21) * (-3) = -15/21 = -5/7

Since the product of the slopes is -5/7 (which is not -1), the diagonals are not perpendicular.

Therefore, the given points (-1, 1), (9, 4), (20, 6), and (10, 1) do not form a rhombus.

As a rhombus requires both sides to be congruent and diagonals to be perpendicular, we cannot proceed to find the area of a rhombus with these points.

To show that the given points (-1,1), (9,4), (20,6), and (10,1) are vertices of a rhombus, we can use the property of a rhombus which states that all sides are equal in length and opposite angles are congruent.

1. Calculate the distance between all pairs of points:
- Distance between (-1,1) and (9,4):
Δx = 9 - (-1) = 10
Δy = 4 - 1 = 3
Distance = (√(Δx^2 + Δy^2)) = (√(10^2 + 3^2)) = √109

- Distance between (9,4) and (20,6):
Δx = 20 - 9 = 11
Δy = 6 - 4 = 2
Distance = (√(Δx^2 + Δy^2)) = (√(11^2 + 2^2)) = √125

- Distance between (20,6) and (10,1):
Δx = 10 - 20 = -10
Δy = 1 - 6 = -5
Distance = (√(Δx^2 + Δy^2)) = (√((-10)^2 + (-5)^2)) = √125

- Distance between (10,1) and (-1,1):
Δx = -1 - 10 = -11
Δy = 1 - 1 = 0
Distance = (√(Δx^2 + Δy^2)) = (√((-11)^2 + 0^2)) = √121

2. Verify that all sides are equal:
- (√109) = (√125) = (√125) = (√121)

Since all four sides have the same length, the given points are vertices of a rhombus.

To find the area of the rhombus, we can consider the distance between the diagonals. The area of a rhombus can be calculated as half the product of the lengths of its diagonals.

3. Calculate the distance between the diagonals:
- Distance between the main diagonal (joining (-1,1) and (20,6)):
Δx = 20 - (-1) = 21
Δy = 6 - 1 = 5
Distance = (√(Δx^2 + Δy^2)) = (√(21^2 + 5^2)) = √466

- Distance between the second diagonal (joining (9,4) and (10,1)):
Δx = 10 - 9 = 1
Δy = 1 - 4 = -3
Distance = (√(Δx^2 + Δy^2)) = (√(1^2 + (-3)^2)) = √10

4. Calculate the area of the rhombus:
Area = (1/2) * (distance between diagonals)
Area = (1/2) * (√466) * (√10)
Area = (√233) * (√5)
Area = (√1165) ≈ 34.14 (rounded to two decimal places)

Therefore, the area of the rhombus formed by the given points is approximately 34.14 square units.