Determine whether parallelogram JKLM with vertices J(-1, -1), K(4, 4), L(9, -1) and M(4, -6) is a rhombus, square, rectangle or all three.

First, plot the parallelogram on graphing paper. Next, calculate the length of each side using the formula for the distance between 2 points (x1, y1), (x2, y2):

((x1-x2)^2 + (y1-y2)^2)^0.5

Then calculate the slope of each line passing through these two points:

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

If the angles are right angles, then the slopes of adjacent sides will be negative inverses of each other. If the slopes of adjacent sides are not negative inverses of each other, then the sides do not meet at right angles.

You should be able to use all this information to determine whether parallelogram JKLM is a rhombus, square, rectangle, or all three

To determine whether parallelogram JKLM is a rhombus, square, or rectangle, we need to examine the properties of these quadrilaterals.

1. Rhombus: A rhombus is a parallelogram with all sides of equal length.
2. Square: A square is a parallelogram with all sides of equal length and all angles equal to 90 degrees.
3. Rectangle: A rectangle is a parallelogram with all angles equal to 90 degrees.

To determine if JKLM is a rhombus, we need to compare the lengths of its sides.

The distance formula is used to find the lengths of the sides:

1. Side JK:
Length = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((4 - (-1))^2 + (4 - (-1))^2)
= sqrt(5^2 + 5^2)
= sqrt(50) = 5√2

2. Side KL:
Length = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((9 - 4)^2 + (-1 - 4)^2)
= sqrt(5^2 + (-5)^2)
= sqrt(50) = 5√2

3. Side LM:
Length = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((4 - 9)^2 + (-6 - (-1))^2)
= sqrt((-5)^2 + (-5)^2)
= sqrt(50) = 5√2

4. Side MJ:
Length = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-1 - 4)^2 + (-1 - 4)^2)
= sqrt((-5)^2 + (-5)^2)
= sqrt(50) = 5√2

From the calculations above, we can see that all sides of parallelogram JKLM have the same length, 5√2. Therefore, JKLM is a rhombus.

Now, let's check if JKLM is also a rectangle:

To determine if JKLM is a rectangle, we need to examine the angles.

1. Angle JKL:
The slope of line JK = (4 - (-1)) / (4 - (-1)) = 5/5 = 1
The slope of line KL = (9 - 4) / (-1 - 4) = 5/-5 = -1
Since the slopes are negative reciprocals, we can conclude that angle JKL = 90 degrees.

2. Angle KLM:
The slope of line KL = (9 - 4) / (-1 - 4) = 5/-5 = -1
The slope of line LM = (4 - 9) / (-6 - (-1)) = -5/(-5) = 1
Since the slopes are negative reciprocals, we can conclude that angle KLM = 90 degrees.

3. Angle LMJ:
The slope of line LM = (4 - 9) / (-6 - (-1)) = -5/(-5) = 1
The slope of line MJ = (-1 - 4) / (-1 - 4) = -5/-5 = 1
Since the slopes are equal, we can conclude that angle LMJ = 90 degrees.

4. Angle MKJ:
The slope of line MJ = (-1 - 4) / (-1 - 4) = -5/-5 = 1
The slope of line JK = (4 - (-1)) / (4 - (-1)) = 5/5 = 1
Since the slopes are equal, we can conclude that angle MKJ = 90 degrees.

From the calculations above, we can see that all angles of parallelogram JKLM are 90 degrees. Therefore, JKLM is a rectangle.

In conclusion, parallelogram JKLM is both a rhombus and a rectangle. However, it is not a square since it does not have all angles equal to 90 degrees.

To determine whether parallelogram JKLM is a rhombus, square, rectangle, or all three, we can analyze its properties.

1. Rhombus:
- A rhombus is a parallelogram with all sides of equal length.
- To check if JKLM is a rhombus, we need to calculate the lengths of all four sides and see if they are equal.

Using the distance formula, we can calculate the lengths of the sides:

Side JK:
d(J, K) = √[(x2 - x1)^2 + (y2 - y1)^2]
d(J, K) = √[(4 - (-1))^2 + (4 - (-1))^2]
d(J, K) = √[5^2 + 5^2]
d(J, K) = √[25 + 25]
d(J, K) = √50

Side KL:
d(K, L) = √[(x2 - x1)^2 + (y2 - y1)^2]
d(K, L) = √[(9 - 4)^2 + (-1 - 4)^2]
d(K, L) = √[5^2 + (-5)^2]
d(K, L) = √[25 + 25]
d(K, L) = √50

Side LM:
d(L, M) = √[(x2 - x1)^2 + (y2 - y1)^2]
d(L, M) = √[(4 - 9)^2 + (-6 - (-1))^2]
d(L, M) = √[(-5)^2 + (-5)^2]
d(L, M) = √[25 + 25]
d(L, M) = √50

Side MJ:
d(M, J) = √[(x2 - x1)^2 + (y2 - y1)^2]
d(M, J) = √[(-1 - 4)^2 + (-1 - 4)^2]
d(M, J) = √[(-5)^2 + (-5)^2]
d(M, J) = √[25 + 25]
d(M, J) = √50

Since all four sides have the same length (√50), parallelogram JKLM is a rhombus.

2. Square:
- A square is a rhombus with all interior angles measuring 90 degrees (right angles).
- To determine if JKLM is a square, we need to calculate the slopes of its sides and check if the adjacent sides are perpendicular (product of their slopes equals -1).

Slope of side JK:
m(JK) = (y2 - y1)/(x2 - x1)
m(JK) = (4 - (-1))/(4 - (-1))
m(JK) = 5/5
m(JK) = 1

Slope of side KL:
m(KL) = (y2 - y1)/(x2 - x1)
m(KL) = (-1 - 4)/(9 - 4)
m(KL) = -5/5
m(KL) = -1

Slope of side LM:
m(LM) = (y2 - y1)/(x2 - x1)
m(LM) = (-6 - (-1))/(4 - 9)
m(LM) = -5/-5
m(LM) = 1

Slope of side MJ:
m(MJ) = (y2 - y1)/(x2 - x1)
m(MJ) = (-1 - 4)/(-1 - 4)
m(MJ) = -5/-5
m(MJ) = 1

The product of the slopes of adjacent sides is 1 * (-1) = -1, which indicates that the adjacent sides are perpendicular. Therefore, parallelogram JKLM is also a square.

3. Rectangle:
- A rectangle is a parallelogram with all interior angles measuring 90 degrees (right angles) and opposite sides of equal length.
- Since JKLM has already been proven to be a square, which has all angles measuring 90 degrees and all sides equal in length, it is also a rectangle.

In conclusion, parallelogram JKLM is a rhombus, a square, and a rectangle.