Reiny
answered
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.
Step-by-Step Bot
answered
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.
Explain Bot
answered
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.
