I'm very lost on all of these, I have a general idea of the postulates/theorems/properties, but I don't know how to go about answering these problems. I've done 1-8 so far. I need the rest.

Accepted Solution

Answer:   9. rhombus   10. rhombus, rectangle, square   11. rhombus   12. rhombusStep-by-step explanation:For classifying quadrilaterals from a list of vertices, it is convenient to consider the diagonals. The diagonals of a rhombus or rectangle will bisect each other. Those of a rhombus will be perpendicular. Those of a rectangle will be the same length. If they are perpendicular and the same length, the figure is a square.Of course, you know that any square is also a rectangle and a rhombus.The diagonals in a list of quadrilateral vertices have endpoints that are alternate pairs of the vertices. It is convenient to look at the coordinate differences.__9. BF = F - B = (12, -2) -(-9, 1) = (21, -3)   EG = G - E = (1, -4) -(2, 3) = (-1, -7)These diagonals are clearly different length, so the figure must be a rhombus.__10. BF = (1, -9) -(1, 3) = (0, -12)  EG = (-5, -3) -(7, -3) = (-12, 0)These diagonals are clearly the same length, so the figure is at least a rectangle. BF is a vertical line (no change in x), and EG is a horizontal line (no change in y), so the diagonals are perpendicular. That makes the figure both a rhombus and a square.__11. BF = (-2, -1) -(-4, -5) = (2, 4)   EG = (-7, -1) -(1, -5) = (-8, 4)These diagonals are clearly different length, so the figure must be a rhombus.__12. The 6-fold symmetry of the figure tells you the central angles are 60°. The quadrilaterals are all parallelograms, so their obtuse angles are the supplement of that, 120°. The symmetry tells you the quadrilateral is a rhombus._____For determining the length of a line segment, the distance formula is ...   length = √((∆x)² + (∆y)²)The segments we computed above are in the form (∆x, ∆y), so it is relatively easy to tell if the sum of squares of those numbers will be the same or different (lengths the same or different).__For determining perpendicularity, two segments are perpendicular when ...   (∆x1, ∆y1) ⊥ (∆x2, ∆y2) if and only if (∆x1)(∆x2) + (∆y1)(∆y2) = 0In vector terms, this sum of products is called the "scalar product" or "dot product" of the two vectors.The mental math required to determine that the pairs of diagonals in the above problems are all perpendicular is not difficult:(21)(-1) + (-3)(-7) = -21 +21 = 0(0)(-12) + (-12)(0) = 0 + 0 = 0(2)(-8) + (4)(4) = -16 +16 = 0_____The attached graph shows the diagonals of the square because we had the geometry app confirm they were the same length.