Provide an appropriate response.
-Are the points A(3, 6), B(6, 9), C( 8, 1), and D(11, 4) the vertices of a parallelogram (opposite sides equal in length)? of a rhombus (all sides equal in length)?
A) yes; yes
B) yes; no
C) no; yes
D) no; no

Question

Quizplus · Accepted Answer

To determine if the points form a parallelogram, we calculate the distances between opposite pairs of points to see if they are equal. For a rhombus, all sides must be of equal length. - AB and CD are opposite sides. The distance between A and B (using the distance formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$) is $\sqrt{(6-3)^2 + (9-6)^2} = \sqrt{9 + 9} = \sqrt{18}$. The distance between C and D is $\sqrt{(11-8)^2 + (4-1)^2} = \sqrt{9 + 9} = \sqrt{18}$.- BC and AD are the other pair of opposite sides. The distance between B and C is $\sqrt{(8-6)^2 + (1-9)^2} = \sqrt{4 + 64} = \sqrt{68}$. The distance between A and D is $\sqrt{(11-3)^2 + (4-6)^2} = \sqrt{64 + 4} = \sqrt{68}$.Since opposite sides are equal in length, the points form a parallelogram. However, since not all sides are of equal length (AB and CD are not equal to BC and AD), the points do not form a rhombus.

Provide an Appropriate Response