quadrilateral RSTU has vertices R(-6,-3), S(3,3), and T(4,-1). what are the coordinates of vertex U if RSTU is a parallelogram?

Well, to find the coordinates of vertex U, we can use the fact that opposite sides of a parallelogram are parallel and congruent. We'll start by finding the equation of the line passing through RS (the side opposite UT), and then find the point of intersection of that line with line TU.

First, let's find the slope of RS:
mRS = (y2 - y1) / (x2 - x1)
= (3 - (-3)) / (3 - (-6))
= 6 / 9
= 2/3

Now, let's find the equation of the line passing through RS using the point-slope form:
y - y1 = m(x - x1)
y - 3 = (2/3)(x - 3)

Next, let's find the equation of the line passing through TU:
mTU = (y2 - y1) / (x2 - x1)
= (-1 - 3) / (4 - 3)
= -4 / 1
= -4

Now, let's find the point of intersection of the two lines:
(2/3)(x - 3) = -4
2x - 6 = -12
2x = -6
x = -3

Substituting the value of x into either equation, let's find the value of y:
y - 3 = (2/3)(-3 - 3)
y - 3 = (2/3)(-6)
y - 3 = -4
y = -1

So, the coordinates of vertex U are (-3, -1). Voilà!

To find the coordinates of vertex U, we need to use the properties of a parallelogram. In a parallelogram, opposite sides are parallel and equal in length.

First, let's find the distance between points R and S using the distance formula:

Distance between two points (x₁, y₁) and (x₂, y₂) = √((x₂ - x₁)² + (y₂ - y₁)²)

Distance between R(-6,-3) and S(3,3):
√((3 - (-6))² + (3 - (-3))²)
= √((3 + 6)² + (3 + 3)²)
= √(9² + 6²)
= √(81 + 36)
= √117
= 10.82 (approx)

Since opposite sides of a parallelogram are equal in length, the distance between S and U must also be 10.82.

Now, let's find the vector between the points R and S. A vector represents the direction and magnitude of a line segment.

Vector v = (x₂ - x₁, y₂ - y₁)

Vector v = (3 - (-6), 3 - (-3))
= (9, 6)

Since the opposite sides of a parallelogram are parallel, vector U must be the result of starting from S and extending the vector v in the same direction.

Using the vector U, we can find the coordinates of U by adding the vector v to the coordinates of point S.

Coordinates of U = (xₛ + vᵢ, yₛ + vⱼ)

Where xₛ and yₛ are the coordinates of S, and vᵢ and vⱼ are the components of vector v.

Coordinates of U = (3 + 9, 3 + 6)
= (12, 9)

Therefore, the coordinates of vertex U are (12, 9) if RSTU is a parallelogram.

Which is the best classification for RSTU?

we need the sides in pairs to have equal slope and equal length.

Slopes:
RS=2/3
ST=-4
So, we need TU to have slope 2/3 and UR to have slope -4.

Since UR goes through (-6,-3), its equation is

y+3 = -4(x+6)
and TU is
y+1 = 2/3 (x-4)

These lines intersect at U=(-5,-7)

Check the lengths:
RS = √((3+6)^2+(3+3)^2) = √117
ST = √((4-3)^2+(-1-3)^2) = √17
TU = √((-5-4)^2+(-7+1)^2) = √117
UR = √((-5+6)^2+(-7+3)^2) = √17

So, it looks like we're OK.