Determine whether the figure with vertices R (-2, 5); O( 1,3); M(-3, -4); Y(-6,-2) is a

parallelogram. Use the Distance and Slope Formulas.

well, the slope of RO is -2/3

If the slope of OM is 3/2, then the two are perpendicular

Or, if
RM^2 = RO^2 + OM^2
then angle O is a right angle.

To determine whether the figure with vertices R(-2, 5), O(1, 3), M(-3, -4), and Y(-6, -2) is a parallelogram, we can use the distance formula and the slope formula.

Step 1: Find the lengths of all sides.
To find the lengths of the sides, we can use the distance formula, which is given by:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Side RO:
Distance = √((1 - (-2))^2 + (3 - 5)^2)
= √(3^2 + (-2)^2)
= √(9 + 4)
= √(13)

Side OM:
Distance = √((-3 - 1)^2 + (-4 - 3)^2)
= √((-4)^2 + (-7)^2)
= √(16 + 49)
= √(65)

Side MY:
Distance = √((-6 - (-3))^2 + (-2 - (-4))^2)
= √((-6 + 3)^2 + (-2 + 4)^2)
= √((-3)^2 + 2^2)
= √(9 + 4)
= √(13)

Side YR:
Distance = √((-6 - (-2))^2 + (-2 - 5)^2)
= √((-6 + 2)^2 + (-2 - 5)^2)
= √((-4)^2 + (-7)^2)
= √(16 + 49)
= √(65)

Step 2: Check if opposite sides are congruent.
If the opposite sides of a quadrilateral are congruent, then it is a parallelogram. Therefore, we compare the lengths of opposite sides:

RO = √(13)
MY = √(13)

OM = √(65)
YR = √(65)

Since the opposite sides are congruent, RO ≈ MY and OM ≈ YR, we can conclude that the figure with vertices R(-2, 5), O(1, 3), M(-3, -4), and Y(-6, -2) is a parallelogram.

Step 3: Check if opposite sides have the same slope.
If the opposite sides of a quadrilateral have the same slope, then it is a parallelogram. Therefore, we calculate the slopes of opposite sides:

Slope of RO:
Slope = (3 - 5) / (1 - (-2))
= -2 / 3

Slope of MY:
Slope = (-2 - 5) / (-6 - (-3))
= -7 / -3
= 7 / 3

Slope of OM:
Slope = (-4 - 3) / (-3 - 1)
= -7 / -4
= 7 / 4

Slope of YR:
Slope = (-2 - (-4)) / (-6 - (-2))
= 2 / -4
= -1 / 2

Since the opposite sides have the same slopes (RO ≈ MY and OM ≈ YR), we can conclude that the figure is a parallelogram.

Therefore, using the distance formula and the slope formula, we have determined that the figure with vertices R(-2, 5), O(1, 3), M(-3, -4), and Y(-6, -2) is indeed a parallelogram.