Provide an appropriate response.
-Are the points A(1, 6), B(4, 9), C( 6, 1), and D(9, 4) the vertices of a parallelogram (opposite sides equal in length)? of a rhombus (all sides equal in length)?
A) yes; yes
B) no; yes
C) no; no
D) yes; no

Question

Quizplus · Accepted Answer

To determine if these points form a parallelogram or a rhombus, we calculate the distances between consecutive points (sides) and diagonals. For a parallelogram, opposite sides must be equal in length. For a rhombus, all sides must be equal. Calculating distances: AB = sqrt((4-1)^2 + (9-6)^2) = sqrt(9+9) = sqrt(18), BC = sqrt((6-4)^2 + (1-9)^2) = sqrt(4+64) = sqrt(68), CD = sqrt((9-6)^2 + (4-1)^2) = sqrt(9+9) = sqrt(18), DA = sqrt((1-9)^2 + (6-4)^2) = sqrt(64+4) = sqrt(68). Opposite sides AB and CD are equal, as are BC and DA, forming a parallelogram. However, not all sides are equal, so it's not a rhombus.

Provide an Appropriate Response