Question

I will prove that quadrilateral STUV is a rectangle by demonstrating that
all sides are of equal measure AND adjacent sides are perpendicular
.

To prove all sides are equal:
ST =
TU =
UV =
VS =

sides are of
measure.

To prove adjacent sides are perpendicular:
slope of ST =
slope of TU =
slope of UV =
slope of VS =
The slopes of any pair of
sides are
.
That being the case, those sides are
.

Therefore, as a result of these two things taken together, the quadrilateral is a
.

Answer 1

I don't see the coordinates or diagram for S, T, U, V. Could you paste the coordinates (or an image) so I can compute the distances and slopes and fill the blanks exactly?

Meanwhile, here is a ready-to-use template and instructions you can use once you have the coordinates:

To prove all sides are equal:
- ST = sqrt[(x_T − x_S)^2 + (y_T − y_S)^2]
- TU = sqrt[(x_U − x_T)^2 + (y_U − y_T)^2]
- UV = sqrt[(x_V − x_U)^2 + (y_V − y_U)^2]
- VS = sqrt[(x_S − x_V)^2 + (y_S − y_V)^2]

If those four distance values are equal, then the sides are all of equal measure.

To prove adjacent sides are perpendicular:
- slope of ST = (y_T − y_S) / (x_T − x_S)
- slope of TU = (y_U − y_T) / (x_U − x_T)
- slope of UV = (y_V − y_U) / (x_V − x_U)
- slope of VS = (y_S − y_V) / (x_S − x_V)

If the product of the slopes of any pair of adjacent sides (for example, slope(ST) · slope(TU)) equals −1, those sides are perpendicular. Do this check for each adjacent pair: ST & TU, TU & UV, UV & VS, VS & ST.

Then conclude:
- Since all four sides are equal in length and each adjacent pair of sides is perpendicular, the figure is a square. A square is a special case of a rectangle, so STUV is a rectangle.

If you give the actual coordinates of S, T, U, V I will compute the distances and slopes and fill in the blank lines numerically.