Tell whether the points are the vertices of a parallelogram (that is not a rectangle), a rectangle, or neither.
-$(-12,-10),(-10,0),(3,7),(1,-3)$
A) Neither
B) Parallelogram
C) Rectangle

Question

Quizplus · Accepted Answer

To determine if the given points form a parallelogram, rectangle, or neither, we can calculate the distances between the points (to check for equal opposite sides) and the slopes of the lines connecting the points (to check for parallel and perpendicular lines). For a rectangle, we need both pairs of opposite sides to be equal and all angles to be 90 degrees (which means the slopes of perpendicular lines are negative reciprocals). For a parallelogram, we only need both pairs of opposite sides to be equal and parallel (equal slopes).1. Calculate distances between points to check for equal opposite sides:   - Distance between (-12,-10) and (-10,0) = sqrt((−10 + 12)^2 + (0 + 10)^2) = sqrt(4 + 100) = sqrt(104)   - Distance between (3,7) and (1,-3) = sqrt((1 - 3)^2 + (-3 - 7)^2) = sqrt(4 + 100) = sqrt(104)   - Distance between (-12,-10) and (1,-3) = sqrt((1 + 12)^2 + (-3 + 10)^2) = sqrt(169 + 49) = sqrt(218)   - Distance between (-10,0) and (3,7) = sqrt((3 + 10)^2 + (7 - 0)^2) = sqrt(169 + 49) = sqrt(218)Since the distances between opposite sides are equal, this is a characteristic of a parallelogram.2. Calculate slopes to check for parallel sides (and perpendicularity for rectangles):   - Slope between (-12,-10) and (-10,0) = (0 + 10) / (-10 + 12) = 10 / 2 = 5   - Slope between (3,7) and (1,-3) = (-3 - 7) / (1 - 3) = -10 / -2 = 5   - Slope between (-12,-10) and (1,-3) = (-3 + 10) / (1 + 12) = 7 / 13   - Slope between (-10,0) and (3,7) = (7 - 0) / (3 + 10) = 7 / 13The opposite sides have equal slopes, meaning they are parallel, which confirms the shape is a parallelogram. To be a rectangle, we would need the product of the slopes of adjacent sides to be -1 (indicating perpendicularity), which is not the case here.Therefore, the points form a parallelogram that is not a rectangle.

Tell Whether the Points Are the Vertices of a Parallelogram (−12,−10),(−10,0),(3,7),(1,−3)(-12,-10),(-10,0),(3,7),(1,-3)(−12,−10),(−10,0),(3,7),(1,−3)

Tell Whether the Points Are the Vertices of a Parallelogram $(-12,-10),(-10,0),(3,7),(1,-3)$